5.3.1 Individual Multidimensional Intertemporal Poverty Index

The objective of this subsection is to develop an analytical formulation of the block approach to the quantification of intertemporal poverty for any arbitrary individual . Essentials to the approach are the individual poverty profiles across periods. It is assumed that individual poverty indicators are observed in each of the periods under consideration. For each person , the per-period poverty profile is a vector , the nonnegative part of the T-dimensional Euclidean space. We can as well say that is person i's intertemporal poverty profile of length . For any given period , represents person i's overall deprivation in the period, as determined by a nonconstant subgroup-decomposable multidimensional poverty index . More precisely, for any , . That is, for any and , gives the extent of poverty corresponding to and the threshold vector . It takes on the minimal value 0 if person is nondeprived in all the dimensions. We assume subgroup decomposability at the outset because of its appealing policy applications.

By assumption,

5.1

The index is ratio-scale invariant, strongly (hence weakly) focused, symmetric, population replication invariant, monotonic, monotonically sensitive, dimensionally monotonic, increasing under a Pigou–Dalton bundle of regressive transfers between two poor persons, increasing (respectively, decreasing) under a correlation-increasing switch if the underlying dimensions are substitutes (respectively, complements) and bounded. It varies continuously with respect to changes in achievement levels of the individual in the period provided that the person's poverty status remains unchanged in the period (period-restricted continuity). (See Chapter 3, for details.)

The individual poverty function considered next turns out to be satisfactory from these perspectives. It corresponds to a member of the Bourguignon–Chakravarty family and is formally defined as

5.2

where and are parameters. Some restrictions on inequality between and have to be imposed for sensitivity under a correlation-increasing switch (see Chapter 3). For all , means that individual is nonpoor in period , whereas a positive value of indicates that he is in poverty in this period.

It may often be necessary to compare intertemporal poverty profiles of different lengths. This type of comparison can be made if we assume that the individual intertemporal poverty index entertains time replication invariance, that is, it remains invariant under replication of the poverty profile with respect to time (see Shorrocks, 2009a). Fulfillment of this property also makes comparisons of individual intertemporal poverty meaningful when possibilities of profiles with different lengths, coming from populations with different sizes, are allowed. To state this principle, formally, we need to define an individual intertemporal index rigorously.

Let , where is the set of natural numbers. Here represents the set of person intertemporal poverty profiles of all possible lengths.

For any , , determines the extent of intertemporal poverty suffered by person over the time profile . By defining over the set all poverty profiles of different lengths, we consider the possibility of allowing comparisons of intertemporal poverty profiles with different lengths using the metric .

The time replication principle may now be formally stated as follows:

1. Time Replication Principle: The intertemporal poverty index is said to be time replication invariant if for all , , ,where is the fold replication of , and is any integer.

In the replicated poverty profile , each entry of is repeated times so that has a length of . In consequence, . Satisfaction of the time replication principle by an individual intertemporal poverty index means that intertemporal poverty is an average concept.

We now suggest two basic postulates for an individual intertemporal poverty index. The first property, monotonicity, involves a curtailment in a deprived dimension's achievement of the person in a period in which he is poor.

In Definition 5.2, person is poor in period because of his positive deprivation in dimension in the profile in period , and his deprivation in the dimension in the same period is higher in the profile . (Recall that we follow the union method of identification of the poor.) His achievements in all other dimensions in and also in all the dimensions in all other periods across the profiles remain unchanged. In consequence, should indicate a higher level of intertemporal poverty than . This postulate, which may be treated as a multidimensional translation of the axiom “monotonicity in outcomes,” suggested by Calvo and Dercon (2009), can now be formally stated as follows.

1. Period-Restricted Monotonicity: For all , and , , given that for all , and , where is an individual poverty function, and is obtained from by a simple reduction in the achievement in a deprived dimension of person .

This axiom demands that a cutback in a deprived dimension's achievement of a person in any period leads to an augmentation in intertemporal poverty level of the person. To understand the axiom in greater detail, let and . The three dimensions of well-being are daily energy consumption in calories by an adult male, per capita income, and life expectancy. The vector of respective thresholds limits is . The achievement profiles of person in the 3 periods are, respectively, , , and . In periods 1 and 2, the person is deprived in life expectancy and income. In period 3, he is deprived only in calorie consumption. In period 1, if life expectancy of the person goes down to 59, then the profile is obtained from the profile by a simple reduction in person i's life expectancy, a deprived dimension of the person, in period 1. The state-restricted monotonicity axiom demands that the former profile should be more intertemporally poverty stricken than the latter.

For and considered in Definition 5.2, assume that is such and . Define as follows: (i) , (ii) for all , and (iii) for all . Then the shortfall of achievement in dimension between the profiles and in period is identical to that between and . But in , person achievement in the dimension is lower than that in the original profile . One may argue that intertemporal poverty should hit a person harder if achievement loss takes at a lower level. To state this formally in terms of an axiom, for all , let , , and , where is an individual poverty function. Then the following postulate, which may be treated as a multidimensional twin of Calvo and Dercon's (2009) “increasing cost of hardship” axiom in the one-dimensional case, may be stated as follows:

1. Period-Restricted Monotonicity Sensitivity: For all , and , .

In the aforementioned example, suppose that person i's life expectancy in period 1 reduces further from 59 to 58.5. Then the period-restricted monotonicity sensitivity axiom appeals that the escalation in the person's intertemporal poverty when the intertemporal achievement profile modifies from to is lower than that when it shifts from to .

In the block approach, aggregation of the components of the individual per-period poverty profile into an individual intertemporal poverty index assumes, under ceteris paribus conditions, that longer breaks between poverty blocks will decrease intertemporal poverty. To understand this requirement, consider the per-period poverty profiles and . There are two breaks in each profile, but the first break in the second profile has higher length. This suggests that the second profile should have lower intertemporal poverty. More precisely, both the profiles are portrayed by a two-period block with poverty levels and , respectively, one one-period block with a poverty value of , and also one one-period block having a poverty value of . But while in the former profile, there is one zero-poverty block between the sequences and , in the latter profile, the same sequences are separated by a zero-poverty block of size 2. In order to make the presence of zero-poverty block significant in a situation of this type, from antipoverty perspective, it is desirable that the former profile should indicate higher individual intertemporal poverty than the latter, under ceteris paribus conditions. We refer to this property as sensitivity to the length in poverty break.

The second feature of the aggregation is that intertemporal poverty increases as the longer length of a poverty block increases at the expense of shorter length of a poverty block. To illustrate this, consider the profiles and . In both the profiles, the positive poverty values , , and appear respectively twice, once, and once. In the first profile, the length of the first poverty block is 3 and that of the second poverty block is 1. Now, in the second profile, the length of the first poverty block increases to 4 and that of the second poverty block reduces to 0. Then the former profile should have less intertemporal poverty than the latter under ceteris paribus conditions. This property reflects sensitivity to the length of poverty blocks.

In sum, we have argued that evaluation of poverty over time from a multidimensional perspective is highly positively correlated with the lengths of individual poverty spells and negatively correlated with the lengths of breaks between two poverty spells.

Now, for any such that , let be the maximal number of consecutive periods including with positive (respectively, zero) per-period poverty values. For instance, let . Since the individual is in poverty in the first two periods, . This is followed by a zero-poverty block of length 1 and hence, . The individual is again in poverty in the next period and out of poverty then, which in turn implies that . The final period in which the person is in poverty is a single period after the fifth block with a zero poverty. This implies that .

The Bossert et al. (2012) multidimensional intertemporal poverty index for person can now be defined as

5.3

Note that nonnegativity of ensures that , where the lower bound 0 is achieved if nobody is deprived in any period ( for all triplets ). This time replication individual intertemporal index unambiguously reduces when profiles have longer breaks between poverty blocks and increases when the length of a poverty block increases. Now, each component of is the individual function, evaluated using person i's achievements, associated with a subgroup-decomposable multidimensional poverty index . Given that are positive, all postulates of are periodwise satisfied by . In the discussion on sensitivity to length in poverty break, the two profiles, we have considered, are of lengths 5 and 7, respectively. The time replication invariance postulate empowers us to compare them with respect to intertemporal poverty.

To illustrate the formula in (5.3), we now calculate its values for the examples considered earlier. The values of person i's intertemporal poverty, as determined by , for these profiles are given as follows:

The first two calculations clearly indicate that individual intertemporal poverty decreases as the breaks between poverty blocks become longer, and the other two calculations show that it increases whenever the lengths of poverty blocks increase.

In the property that reflects sensitivity of to the length of a poverty block, we assumed an arbitrary number of poverty periods. We will now state a related property of a general individual intertemporal poverty index involving the lengths of poverty blocks in profiles with only two poverty periods. This postulate will be stated as a necessary and sufficient requirement. To illustrate the idea, consider the profile . The common length of each of the two poverty blocks in this profile is 1, and they are separated by a nonpoverty block of length 2. Now, if we swap the poverty experiences of periods 3 and 5 in this profile, the resulting profile turns out to be . The value of the index for this postswitch intertemporal profile is . But . This means that contiguous locations of poverty occurrence increase intertemporal poverty. In other words, contiguous poverty periods in an intertemporal poverty profile generate higher impact on . Conversely, we can derive the profile from the profile by moving two contiguous periods of poverty further apart. This movement decreases the value of . Loosely speaking, this example shows that, in this particular case, switch of atemporal poverty locations in an intertemporal profile increases intertemporal poverty if and only if their postswitch positions are contiguous. We can state this property of for a general intertemporal poverty index as follows:

1. Block Monotonicity: Suppose that there are only two distinct and nonadjacent poverty periods in the intertemporal poverty profile , where and . Now, if is obtained from by trading temporal positions of these two poverty incidents to and , where , then holds if and only if .

Since contiguous sites of poverty experiences increase the length of a poverty block, which in turn intensifies intertemporal poverty, we refer to this axiom as block monotonicity axiom. Hoy and Zheng (2015) defined intertemporal poverty on the space of lifetime income distributions and called it strong chronic poverty axiom.

Of the four properties we introduce next, for a general index , the first three were suggested by Bossert et al. (2012), and the fourth one is by Calvo and Dercon (2009). The first postulate claims that if there is only one period, then one-period poverty is the same as individual intertemporal poverty. Formally,

1. One-Period Equivalence: For all ,
   5.4

The next property says that when the individual is in poverty for all the periods, then for any subperiod , , intertemporal poverty can be calculated as the sum of the intertemporal poverties of the profiles and . More precisely,

1. Single-Block Additive Decomposability: For all , all and all ,
   5.5

   where is the strictly positive part of the dimensional Euclidean space.

Repeated application of this postulate shows that

5.6

given that for all . That is, an individual's intertemporal poverty is simply the sum of periodwise poverty levels. This decomposition becomes quite useful from antipoverty policy perspective; it enables us to identify those periods that are more poverty stricken. In consequence, if , we can partition the set of periods into nonoverlapping subsets, say, and , and intertemporal poverty equals .

According to the third property, for all nonnegative poverty profiles, if two subgroups of periods are separated by at least one zero-poverty period, then the total intertemporal poverty can be expressed as the weighted average of subgroup intertemporal poverty values, where the weights are the proportional lengths of the subgroups. Analytically,

1. Across-Blocks Average Decomposability: For all , all , all if or , then
   5.7

This axiom shows significance of the lengths of poverty blocks and lengths of blocks out of poverty. The length of a block becomes an important characteristic for evaluation of intertemporal poverty. Accordingly, if , then equals .

However, across-blocks average decomposability does not apply to any arbitrary poverty profile, rather to some restricted profiles. A stronger decomposability condition, presented by Calvo and Dercon (2009), applies to any arbitrary profile of poverty. This stronger form, the subperiod decomposability postulate, does not require the separating spell or to be a zero-poverty spell, where is arbitrary. More concretely,

1. Subperiod Decomposability: For all , all , all ,
   5.8

This is a direct counterpart of the population subgroup decomposability axiom considered in the poverty measurement literature. Repeated application of the postulate shows that (see also Hoy and Zheng, 2015).

The following axiom also specifies a reasonable requirement.

1. Intertemporal Continuity: For all , , is a continuous function of each period's poverty level provided that the person's poverty status remains unchanged in all periods.

The poverty statistic is a violator of subperiod decomposability. However, it fulfills the other three postulates of the set of the aforementioned four postulates. In the example taken earlier, , , and . Then the percentage contribution made by the subgroup of the periods to total intertemporal poverty of the person is . Hence from antiperiod policy perspective, the subgroup of periods requires more attention. This illustration demonstrates an appropriate policy relevance of the axiom across-blocks average decomposability.

We now employ the periodwise Bourguignon–Chakravarty poverty function for individual in (5.3). The resulting individual intertemporal poverty index becomes a positive weighted function of individual temporal poverty levels , where , person poverty in period , is evaluated using his achievements in the period by the individual function associated with .

5.9

If for at least one , then for a given value of , an increase in the value of increases the intertemporal continuous poverty index in (5.9). Since for any , is positive, this index unambiguously verifies the period-restricted monotonicity axioms for any and .

Another desirable postulate is formulated in terms of a transfer of a bundle of achievements from one period to another, where the dimensions that are affected by the transfer are more deprived in the former than in the latter. To state this property formally, let be the identical set of dimensions in which person is deprived in periods and in , where are arbitrary and . Formally, . Assume further that the person's deprivations in different dimensions in period are lower than his corresponding deprivations in period . More precisely, for all , .

Since can at most be , it is ensured that for any , . The inequality guarantees that the regressive transfer in any dimension from the bequest period to the beneficiary period does not permit the person to be nondeprived in the dimension in the beneficiary period.

In the example we have taken earlier to illustrate the monotonicity axioms, let and . Then . Choose . As a result, and . Then we say that the intertemporal profile is obtained from the profile by an across-periods Pigou–Dalton bundle of regressive transfers. (Strictly speaking, since income is the only affected dimension, it is one-dimensional regressive transfer considered by Bourguignon and Chakravarty (2003).)

The following axiom, which can be viewed as a multidimensional companion of the one-dimensional intertemporal regressive transfer axiom advocated by Gradin et al. (2012), can now be stated:

1. Across-Periods Transfer: For all , , given that for all , and , where is an individual poverty function and is obtained from by an across-periods Pigou–Dalton bundle of regressive transfers.

The across-periods transfer axiom requires that individual intertemporal poverty should rise under an across-periods Pigou–Dalton bundle of regressive transfers. Similarly, intertemporal poverty should decrease if a bundle of progressive transfers takes place. The two time locations and we have chosen in the period profile are arbitrary. Therefore, one sufficient condition that ensures satisfaction of the transfer axiom is that all poverty spells in (5.9) are assigned the same weight in the aggregation. For unequally weighted spells, some restrictions may be required for fulfillment of the axiom. Given and , a sufficient condition that ensures its verification of the axiom of across-periods Pigou–Dalton bundle of regressive transfers by in (5.9) is that . An intertemporal poverty index will be called transfer sensitive if it fulfills the across-periods transfer postulate.

5.3.2 Aggregate Multidimensional Intertemporal Poverty Index

We assume that the society intertemporal poverty index is a nonnegative real-valued function of individual intertemporal poverty components. Since under this assumption, the index relies only on individualistic intertemporal poverty features, we name it as independence of irrelevant information.

For the reason that a typical element of the set of all intertemporal poverty profiles of an person society is , the set of all possible intertemporal poverty profiles at the society level is . Following Bossert et al. (2012), we define the average intertemporal poverty index for the society as , where for all , ,

5.10

For any , gives the level of overall intertemporal poverty associated with the intertemporal poverty profile across persons. That is, the overall intertemporal poverty level is the simple arithmetic average of individual intertemporal poverty values. We allow variability of the population size to broaden the framework sufficiently so that across-populations collation of overall intertemporal poverty becomes possible.

For of the form given by (5.9), the resulting index turns out to be

5.11

This population replication invariant overall intertemporal poverty index enables us to perform cross-population comparisons of intertemporal poverty allowing variability of the sampling period. It can as well be employed to compare intertemporal poverty extents of the same population for different sampling periods. The population replication invariance principle views global intertemporal poverty as a per-capita concept. Its subgroup decomposability property facilitates us to button down those persons in the society who are beset more by intertemporal poverty and hence to design antipoverty policy. This policy is about lifetime elude or lack of elude from poverty. It is definitely about a future situation, not just involving one future period distress.

The per-capita notion of global poverty is also reflected by the average critical levels postulate, which necessitates that if a person with average poverty level migrates to the society, then global poverty remains unchanged (see Blackorby et al., 2005). This property is captured by the index .

1. Average Critical Levels: For all , for all , .

Observe that the average of intertemporal poverty levels across persons takes into account all information on periodwise poverty thresholds. In consequence, the choice of the average intertemporal poverty as the critical level does not lead to any loss of information.

If any two individuals trade their positions in (5.10), then this exchange has no effect on because of its anonymity property. It also gladdens an impartiality principle with respect to an increment or a decrement in individual poverty. If a single person's intertemporal poverty changes by a certain amount, then it is immaterial whose poverty changes. For the postulate to be well defined, it is necessary to assume that the change in poverty is the same across persons. In order to state this postulate rigorously, let , where for all and .

1. Incremental Equity: For all , , for all , for all and for all , such that , .

Thus, incremental equity needs that the impact on global poverty of a change in any person j's poverty has the effect of a similar change in a different person k's poverty. Consequently, the persons and are treated anonymously from this perspective.

The index takes on the value 0 if for all . More generally, if for all , then . For any , is bounded between the minimal and maximal values of individualwise intertemporal poverty levels. More precisely, .

If for some pair (j, t), there is an increase in , say, following a reduction in , then increases unambiguously. Formally,

1. Strong Monotonicity: For all , and for all , ,where for some and for all .

This strong monotonicity property of demands that society's intertemporal poverty increases under an upsurge of any individual's intertemporal poverty. It implies its weak sister, which says that overall intertemporal increases when all the individuals' intertemporal poverty levels get augmented. More precisely,

1. Weak Monotonicity: For all , and for all , , where for all .

Among the other notions of monotonicity that are captured by are (i) ratio-scale improvement, (ii) translation-scale improvement, and (iii) minimal increasingness. According to the ratio-scale improvement postulate, an equiproportionate contraction in individualwise intertemporal poverty values leads to a shrinkage of global intertemporal poverty (Shorrocks, 1983). More concretely,

1. Ratio-Scale Improvement: For all , for all ,where , and for all scalars , , .

The translation-scale improvement condition claims that an equal absolute diminution of amounts of all persons' intertermporal poverties generates a lessening of society's intertemporal poverty value (Shorrocks, 1983).

1. Translation-Scale Improvement: For all , for all , , and for all scalars such that for all , .

Finally, minimal increasingness appeals that if intertemporal poverty is equal across individuals, then lower poverty is preferred to higher poverty (Blackorby and Donaldson, 1984). More explicitly,

1. Minimal Increasingness: For all ,and for all , where a > b, .

This axiom, formulated in terms of an equal intertemporal poverty across persons, is very weak and appealing.

5.3.3 A Review of Some Related One-Dimensional Proposals

The literature contains several recommendations for measuring individual intertemporal poverty in the univariate case. A comparative analysis of the proposals that bear similarity with our block-approach-based endorsement explored earlier, when it is adapted to the single-variable situation, will be a useful exercise. This is the objective of this subsection.

We denote individual i's income in period by , , and . Denote the time-invariant income poverty line by . The censored income level associated with is symbolized by , that is, . We write for the income deprivation indicator for the person in period .

The three steps that were employed in the Calvo–Dercon formulation for developing lifetime individual statistic of poverty are as follows: focus (truncation of above-threshold incomes in different periods), transformation (all atemporal deprivations are transformed by some increasing and strictly convex function so that unidimensional monotonicity and monotonicity sensitivity axioms are fulfilled), and aggregation of transformed deprivations into an overall index of individual lifetime poverty. We denote these three steps by F, T, and A, respectively. The six possible orderings of the steps are FTA, FAT, TFA, TAF, AFT, and ATF. Since the orderings FTA and AFT draw out all the insights in TFA and ATF, respectively, the latter two can be ignored.

Under the sequence FTA, after focus, identical increasing and strictly convex transformation is imposed on the atemporal deprivations to preserve the monotonicity axioms, before aggregation. The resulting indices resemble “the well-known Chakravarty and Foster–Greer–Thorbecke measures of aggregate poverty” (Calvo and Dercon, 2009, p. 40). Formally, they are given respectively by

5.12

where , , and

5.13

where and . The common parameter in the aforementioned two indices is a discounting parameter. Consequently, ascertains the rate of time discounting. It gives us the weights assigned to transformed deprivations in different periods over the profile . If the weights are chosen in decreasing order of time, then distant future-period spells are paid lower attention in the aggregation. A necessary condition that guarantees this is that . Thus, while in , equal weight is assigned to the spells in a block, and as the size of the block increases, the constant weight across the spells in the block increases, for the Calvo–Dercon indices, the weight unambiguously decreases as the period becomes more distant. If , each spell is allotted the same weight (=1), and this happens irrespective of its location in the profile . In other words, these indices remain invariant under any rearrangement of the periods in the profile . As a result, there is no discounting of the periods. In this case, a progressive transfer of income from a less deprived period to a more deprived one decreases individual intertemporal poverty. In other words, of two deprived periods, the higher one can be subsidized by the lower one.

The two subperiod decomposable Calvo–Dercon indices comply with the one-dimensional versions of the monotonicity and monotonicity sensitivity axioms. For and , becomes the number of periods in which the person stays below the poverty line. This is the individual intertemporal period count index for income poverty.

When the sequences AFT and TAF are adopted, the variants of the pair of indices ((5.11) (5.12)) are given respectively by the pairs

where the parameters , and obey the same restrictions as in (5.12) and (5.13). Each component of the two pairs is a violator of subperiod decomposability. While each constituent of the first pair fails to meet monotonicity sensitivity, monotonicity is risked for any constituent of the second pair. The set of violators of subperiod decomposability and monotonicity sensitivity also include indices derived under the sequence FAT. In view of these problems identified with the indices underlying these three sequences, we do not analyze them further.

Dutta et al. (2013) proposed a deviant of the Bossert–Chakravarty–D'Ambrosio quantifier of individual intertemporal poverty. They noted that in the Bossert–Chakravarty–D'Ambrosio formulation, the distribution of nonpoverty spells does not play any role in the determination of individual intertemporal poverty. Their alternative approach relies on the assumption that the longer nonpoverty spells experienced by an individual prior to becoming poor make him more capable of dealing with poverty. However, preceding poverty periods cannot be made milder by nonpoverty spells. In other words, affluence cannot weaken a previous poverty spell but can help one to be better equipped to fight poverty in future. They also characterized a family of individual intertemporal poverty indices.

The explicit form of their alternative recommendation is given by

5.14

where is income deprivation indicator in period , is the number of consecutive poverty periods prior to period , is the number of nonpoverty periods immediately prior to period , and , , are positive parameters. The parameter represents sensitivity of period-by-period deprivation indicators to intertemporal poverty in the sense that given values, how a change in changes , and and determine, respectively, sensitivity of the poverty block size prior to period and the number of nonpoverty periods just prior to period . Evidently, under ceteris paribus assumptions, increases as increases (i.e., as poverty periods cluster) and decreases as increases (i.e., as nonpoverty periods cluster just before a poverty period). The approach has a clear merit – in the aggregation, it incorporates the characteristic of appeasing in terms of preceding nonpoverty spells and increasing impact of bunching of poverty spells. This nonnegative individual intertemporal index that upholds the block monotonicity and one-period equivalence axioms is a normalized index in the sense that it takes on the value 0 if for all . Dutta et al. (2013) suggested a modification of (5.14) by replacing by and , if income is above certain threshold limit , and 1 otherwise. Under this modification, the index verifies a weak poverty mitigation axiom, which claims that there is some particular level of income, , exceeding the poverty line, such that income in a nonpoverty period can lighten the effect of poverty episodes in other periods but only up to the specified limit . To understand this, let and be two lifetime income profiles, the poverty line and . Then should have less lifetime poverty than because mitigation effect happens to exist up to the limit . Further, the profile has the same intertemporal poverty as , although period 1 income in is lower than that in . The reason behind this is that an increase in income beyond has no further mitigation effect. However, it fails to verify the strong poverty mitigation axiom, which requires continuity of the mitigation operation and impact to be increasing with income (see Hoy and Zheng, 2015).

Another one-dimensional variant of the Bossert–Chakravarty–D'Ambrosio proposal was suggested by Gradin et al. (2012). According to these authors, a regressive transfer of income from one period to another, where the size of the poverty block to which the bequest period belongs is at least as high as that of the poverty block to which the beneficiary period is attached, should increase individual intertemporal poverty under ceteris paribus assumptions. (See the across-periods transfer axiom presented in the Section 5.3.1.) To perceive this in greater detail, let be person i's intertemporal income profile and be the income poverty line. The person is poor in periods 2,3,4,6, and 7. Now, consider a regressive income transfer of size 1 from period 3 to period 6 generating the profile . The size of the block to which the bequest period pertains is 3 (with poverty periods being 2, 3, and 4). In contrast, the size of the block to which the beneficiary period is attached is 2 (with poverty periods being 6 and 7). Hence, this regressive transfer should increase person i's intertemporal poverty.

The alternative functional form for person i's intertemporal poverty index suggested by these authors is

5.15

where is the individual income poverty function in period and is the same as in equation (5.3). Standard examples of can be and , where and are parameters. These two functions correspond respectively to the Foster et al. (1984) and Chakravarty (1983) poverty indices. The parameter attaches higher weight to longer poverty spells reflecting the idea that continuous enlargement of the size of a poverty block intensifies a person's intertemporal poverty experience. However, is a violator of the time replication invariance axiom. The authors argued that for looking at poverty differences between two intertemporal distributions with sizes and , where , one can choose a subsample of periods of the second distribution and make the necessary comparison. Their index fails to take into account the act of poverty mitigation. The authors also suggested an aggregate index of intertemporal poverty by considering an average of an increasing strictly convex transformation of values over the entire population.

The key factor underlying the Mendola–Busetta path of measuring intertemporal poverty is the “cumulative hardship” property, which requires intensification of intertemporal poverty as periodwise poverty situations become closer even if the postswitch stations of poverty periods are not contiguous. It is weaker than the block monotonicity axiom, which requires augmentation of intertemporal poverty only when postrotated locations of poverty periods are contiguous.

The cumulative hardship postulate is consistent with the Hoy and Zheng (2011, 2015) chronic poverty axiom, which we state next on the space of intertemporal poverty profiles:

1. Chronic Poverty: Suppose that there are only two distinct and nonadjacent poverty periods in the intertemporal poverty profile , where and . Now, if is obtained from by exchanging temporal locations of these two poverty situations to and , where for some , then .

This statement of the axiom has a minor difference with the Hoy–Zheng formulation since we replace their weak inequality by the strict inequality . All indices that verify this axiom are members of the class that can be identified under the category “the closeness approach.” The proposals , , and are violators of this axiom since in this axiom, it is not unambiguously required that the postswitch locations are contiguous.

The Mendola and Busetta (2012) index of individual intertemporal poverty for determining the extent of poverty in a profile spanning periods is defined as

5.16

where is a normalization coefficient and is a decay factor discounting the individual's early period poverty happenings. Although it satisfies the chronic poverty axiom, it is a violator of the decomposability axioms analyzed earlier and the two poverty mitigation axioms.

Essential to the Hoy and Zheng (2011) closeness-reliant proposal is the chronic poverty axiom. According to these authors, each period of poverty and an individual's entire lifetime are important ingredients of an individual's lifetime poverty. They characterized the following functional form of individual intertemporal poverty index using an axiomatic framework:

5.17

where , with being the one-dimensional poverty index; and_. . The term is a weighted average of periodwise poverty experiences of the person, and is the extent of poverty associated with the person's lifetime average income. In the polar case , intertemporal poverty is represented by the person's deprivation arising from the shortfall of his lifetime average income from the threshold limit. Consequently, poverty situations of different periods do not play any role here. In other words, knowledge of period-by-period poverty extents is not necessary. In contrast, when , a recollection period-by-period poverty experience is required to determine the overall intertemporal poverty of the person. Hence, the parameter may be interpreted as representing a “memory factor” in the sense that the extreme cases and correspond, respectively, to “perfect recall” and “no recall” of poverty episodes. A value of lying between 0 and 1 shows that each term of the across-periods poverty sequence and lifetime poverty value are decisive factors in the evaluation of individual intertemporal poverty. In this intermediate case, the index upholds the strong poverty mitigation axiom.

Hoy and Zheng (2011) showed that satisfaction of the early poverty axiom by requires that is a decreasing function of . The early poverty axiom demands that poverty episodes in early phases of life have more harmful effects on life. Poverty in childhood periods is likely to have detrimental effects on physical and mental conditions of a person later in life. Similarly, verification of the chronic poverty axiom by necessitates that is strictly concave in . Examples of the function that meet these requirements are and , where and are constants.

As noted by Zheng (2012), the additive structure of does not enable us to take into account across-periods poverty interactions. To overcome this, Zheng (2012) developed an axiomatic characterization of the following class of gravitational indices of intertemporal poverty

5.18

where is continuous in its arguments, increasing in the first two arguments, decreasing in the third argument, , and . Zheng (2012) refers to this class as gravitational class since its formulation relies on Newton's Universal Law of Gravitation – “the force is proportional to the product of two masses and inversely proportional to the square of distance between the point masses.”

Two illustrative examples of the family (5.18) are

5.19

5.20

where , and are positive constants. As these two examples indicate, is increasing in each pair of poverty experiences, and as the gap between any two poverty spells reduces so that decreases, increases.

5.5.1 A Review of One-Dimensional Duration-Reliant Offers

Recall the notation we have introduced in Section 5.3.3. We denote individual i's censored income in period by and the time-invariant income poverty line by , where and . The income deprivation indicator of person in period is . The duration approach identifies a person as chronically income poor (chronically poor, for short), if he remains in income poverty for at least a certain fraction of time periods . Equivalently, the person is chronically poor, by this method of identification, if he becomes income deprived in at least periods, where , that is, is the minimum number of periods for which holds. It should be clear that the value of is unique. For instance, if we have observations on achievements for 11 periods and , then .

We refer to as the duration threshold. Several suggestions have been made in the literature concerning the choice of . For instance, according to Gaiha and Deolikar (1993), those families with incomes below the poverty line in at least five out of nine periods of observations can be taken as chronically poor. Hulme and Shepherd (2003) argued that the necessary condition for a person to be identified as suffering from chronic capability deprivation is poverty experience for at least five successive years (see Hoy and Zheng, 2015, for further discussion). However, our definition of chronic poverty does not require poverty experiences in consecutive periods. Unless specified, we will assume that is given.

Let be a panel of income distributions of persons over periods. The tth column of the panel gives the distribution of income among persons in period . Similarly, the ith row of the panel shows person i's incomes in periods. Let be the set of chronically poor persons in . In other words, is the set of all the persons who remain in poverty for at least periods of time. Then , the cardinality of the set , gives the number of chronically poor persons in . The fraction is the chronic head-count ratio, the proportion of persons that are chronically poor in .

The chronic income poverty index suggested by Foster (2009) is defined as

5.30

where is a parameter. We can rewrite (5.30) as , the product of the chronic head-count ratio and the average of the transformed deprivations of the chronically poor across all periods. As Foster (2009) mentioned, satisfies several useful properties. For , the index becomes the duration-adjusted head-count ratio , where is the average duration of poverty among the chronically poor, given by .

According to Jalan and Ravallion (1998), a person is regarded as chronically poor if the mean of his periodwise incomes falls below the poverty line. Let be the panel of income distributions in which for any , is replaced by for all , where is the average income of person across periods. In other words, is obtained from by replacing the entries in the ith row by the common number , for all .

The Jalan–Ravallion index of chronic income poverty, , can then be obtained by aggregating the squared deprivations of all chronically poor persons in the smoothed panel . Formally,

5.31

where the set of chronically poor persons now consists of all those persons whose average incomes over the entire length of the period profile fall below the poverty line. Evidently, in this case, the chronic poverty identification problem is a one-period issue: if the mean income of a person over all the periods is below the poverty threshold, then he is chronically poor, otherwise not. This analysis clearly shows that, given the one-period notion of identification of the chronically poor, the Jalan–Ravallion index can be calculated using the type of aggregation invoked in Foster (2009), under the additional assumption that .

The Foster (2009) approach can, in fact, be extended to the entire family of subgroup-decomposable income poverty indices. The general ditto of can be defined as

5.32

where is continuous, decreasing, and strictly convex. For becomes Foster's (2009) proposal. Alternatively, if we choose , then becomes a chronic poverty translation of the Chakravarty (1983) income poverty index (see Chakravarty, 2009). We can also apply the Jalan–Ravallion identification criterion to to get variants of .

5.5.2 Axioms for a Chronic Multidimensional Poverty Quantifier

Throughout this and the next two subsections, we follow the notation adopted in Section 5.2. Given a well-defined method of identification of the multidimensionally poor in each period, the duration approach identifies a person as chronic multidimensionally poor if he remains in poverty for some exogenously given periods of time. As before, we adopt the union method of identification of poor.

This duration-dependent approach involves a two-step identification problem. The first step requires identification of the multidimensionally poor persons in each period. The exercise at the second step is to identify the chronically poor among these multidimensionally poor persons in different periods. Recall our notation that stands for the set of all dimensional intertemporal distribution matrices when the population size is , and denotes the set of such matrices for all possible population sizes (assuming that and are given).

For all , , let be the set of all persons who are chronically poor in , that is, the set of all persons who are counted as multidimensionally poor for at least periods in the intertemporal distribution matrix . Formally, . We denote the number of chronically poor persons in by . A multidimensional chronic poverty index in this framework can be written as a function , where .

As an illustrative example assume that and . The social matrices for the four periods are , , , , and the vector of common poverty cutoffs is . The entries in the first row of matrix indicate person 1's achievements in period , where . Similarly, figures in the other rows of the four matrices can be explained. By the union method of identification of the poor, while person is poor in all the four periods, person 3 is poor only in period 4. On the other hand, person 1 is poor in periods 1 and 4. Assume that the duration threshold . Then although persons 1 and 2 are chronically poor, person 3 is not so. It may be worthy to note that by the intersection identification procedure, persons 1 and 2 are poor in period 1. However, nobody is chronically poor by this notion of identification.

The corresponding intertemporal social matrix with 12 rows and 2 columns, denoted by , is now obtained by placing the rows of , where , contiguously as rows of from above to below. In consequence, the entries in rows 1, 4, 7, and 10 of quantify person 1's achievements in periods 1, 2, 3, and 4, respectively, in the two dimensions. Other rows of can be explained analogously. The number here is 2 and the corresponding set consists of persons 1 and 2. We will use the matrix to illustrate the axioms defined for a general index.

The explicit representation of the intertemporal social matrix , associated with our example, is given by

The following axioms can now be stated for a general multidimensional chronic poverty index . Of these, the first two axioms parallel the Bourguignon–Chakravarty focus axioms. The four axioms that follow these are directly adapted to the chronic poverty setup from the literature. Each of the next five axioms is a multidimensional version of Foster's (2009) corresponding unidimensional postulate. They are followed by a chronic twin of the Alkire and Foster (2011) dimensional monotonicity axiom. The remaining postulates are chronic reproductions of the corresponding properties proposed in Chapter 3.

1. Chronic Weak Focus: For all , , , suppose that person is not chronically poor in , and is obtained from as follows: for a triplet , where and for all triplets . Then .

According to this axiom, an increase in the achievement in a dimension of a chronically nonpoor person in any period does not change the value of the poverty index. That is, the poverty index is independent of the achievements of all persons who are chronically nonpoor. In the matrix , person 3 is chronically nonpoor. An increase in his achievement in dimension 1 in period 4 from 3 to 4 reduces his poverty level in the period but does not affect his chronic poverty status.

1. Chronic Strong Focus: For all , , , suppose that is obtained from as follows: for a triplet , where , is such that and for all triplets . Then .

In this case, person may or may not be chronically poor. In either case, if his achievement in a nondeprived dimension in some period increases, the poverty index remains unaffected. If person is chronically nonpoor, then the strong focus axiom reduces to its weak version. In person 1, who is chronically poor, is nondeprived in dimension 1 in period 3. His chronic poverty position remains unaltered, if this achievement increases from 9 to any higher level.

1. Period-Restricted Anonymity: For all , , , suppose that is obtained from as follows: , where is any permutation matrix, is arbitrary, and for all . Then .

This postulate demands that a rearrangement of the rows of the distribution matrix in any period, keeping the distribution matrices in all other periods unchanged, does not affect chronic poverty. Thus, in our social matrix , if we exchange only the first two rows, that is, the first row now becomes (2,2), whereas the second row is given by (3,3) and all other rows remain unaltered, then the chronic poverty levels of the new intertemporal matrix and the original matrix are the same.

The next axiom becomes helpful in cross-population comparisons of chronic poverty.

1. Chronic Population Replication Invariance: For all , , , ,where is the -fold replication of , being a positive integer, that is, in each row of appears times.

This axiom ensures that chronic poverty is measured in per-capita terms. Consequently, comparison of chronic poverty levels of two societies whose population sizes are different becomes possible. If the intertemporal matrix is replicated twice, in the resulting matrix , each row of appears twice and the postulate requires that .

1. Chronic Strong Ratio-Scale Invariance: For all , , , suppose that is obtained from as follows: for all , , where is any positive diagonal matrix. Then .

According to this axiom, if all the achievements in any dimension in all the periods and the corresponding threshold limit are subjected to the same equiproportionate change, then chronic poverty remains unchanged. A weaker form of this axiom, chronic ratio-scale invariance, demands that the proportionality factor is the same across the dimensions in all the periods. For four matrices associated with , define by and let the corresponding intertemporal matrix be , where . Then the strong invariance postulate demands that .

The postulate, subgroup decomposability, establishes consistency between local and global evaluations of chronic poverty in a specific way.

1. Chronic Subgroup Decomposability: For all , , ,
   

   where and is obtained by placing the matrices from above to below for , where .

The formulation of the axiom indicates that for any division of the population into two or more nonoverlapping subgroups, aggregate chronic poverty is the population share weighted average of its subgroup brothers. Since the numbering of the matrices is arbitrary, the arrangement of the rows of is also arbitrary. Under an alternative numbering of the matrices, the positioning of the rows of will be in the manner consistent with the numbering. Suppose that a reduction in chronic poverty in one subgroup, say, as a result of implementation of some targeted poverty alleviation policy, takes place. Then, under the ceteris paribus condition that poverty extents remain fixed in all other subgroups, this shrinkage in the particular subgroup's poverty must lead to a reduction in overall chronic poverty.

To illustrate this axiom, suppose that the three individuals we have considered in our example have been partitioned into two subgroups and , with respect to their regions of residence, say, where consists of person 1 and the other subgroup consists of persons 2 and 3. Then the axiom claims that ,where and denote the intertemporal social matrices associated with the subgroups and , respectively.³

1. Time Anonymity: For all , , , if the sequence of matrices is obtained by a reordering of the sequence of period-by-period matrices under , then .

This desideratum claims that an interchange of positions two periodwise distribution matrices in the sequence of timings does not affect chronic poverty. Consequently, if the distribution matrix that appears in period under appears in period under , and for the remaining periods the distribution matrices under and are the same, then chronic poverty in will be equal to that in . In the illustrative matrix , suppose that the positions of and are interchanged, and the positions of and are kept unchanged. If we denote the resulting intertemporal distribution matrix by , then the time anonymity axiom asserts that .

Clearly, an intertemporal poverty index that verifies the block monotonicity axiom so that higher weights are assigned to poverty blocks with larger sizes is a violator of this postulate. Evidently, the time anonymity axiom does not make any distinction between two situations with the same number of poverty spells where in the former, the poverty spells appear consecutively, but in the latter, they are separated by at least one zero-poverty spell. It treats the profiles and as identically intertemporally poor. But the block approach regards the latter as poorer than the former because in the former, the lengths of the two poverty blocks are 3 and 1, respectively, whereas in the latter, the single poverty block has a length of 4. The time anonymity axiom brings out a major distinguishing feature between the two approaches. An index that agrees with this axiom also fails to meet the Hoy–Zheng chronic poverty axiom.

1. Chronic Monotonicity: For all , , , suppose that the matrix is obtained from as follows: for a triplet , where person is chronically poor, , , and for all triplets . Then .

If achievement in a deprived dimension of a chronically poor person reduces, then a natural requirement is that poverty should go up. This is what the chronic monotonicity axiom demands. In , if achievement in dimension 2 in period 1 of person 2, a chronically poor person, goes down from 2 to 1, then poverty should indicate an upward trend.

1. Time Monotonicity: For all , , , suppose that person is chronically poor in and is related to as follows: for a triplet , and for all triplets . Then .

Poverty should go up under an increase in the number of periods of deprivation suffered by a chronically poor person in some dimension. Our time monotonicity axiom states this necessity. In the matrix , person 1 is chronically poor but nondeprived in dimension 1 in period 2. If this achievement gets slashed from 7 to 4, then his duration of deprivation in the dimension increases from 2 to 3. Hence, chronic poverty should go up.

1. Duration Monotonicity: For all , , , given the duration threshold , if , where is an integer such that , then .

If the duration threshold goes down, then there is a possibility that some newer persons who were not chronically poor originally become chronic poverty stricken now. Hence, we are adding some new chronically poor persons without changing the statuses of the existing poor. Consequently, chronic poverty should not go down under a reduction in the duration threshold. In our social matrix , suppose that reduces from 2 to 1, then all the three persons become chronically poor, whereas for , only two persons are chronically poor. Accordingly, .

None of the axioms stated so far was concerned with the inequality among the chronically poor. In the case of cross-sectional multidimensional poverty, if there is a Pigou–Dalton bundle of transfers of achievements, in deprived dimensions, from a poorer poor to a richer poor that do not change their statuses, then the posttransfer social distribution should have higher multidimensional poverty than the pretransfer one.

Given , , , , and , suppose that in all the dimensions of , the common set of deprived dimensions of the chronically poor persons and in period , the former has higher deprivations than the latter. More precisely, for all , , where ,with and being arbitrary. Assume that is nonempty.

We have assumed at the outset that the same set of deprived dimensions of the chronically poor persons and in period , with the additional characteristic that has higher deprivations than , is nonempty. Condition (i) of Definition 5.4 demands that all individuals except persons and have identical achievements in all the dimensions in both and . Part (a) of condition (ii) says that each of persons and has identical achievements in all nondeprived dimensions in period . According to part (b) of condition (ii), we get and by dimensionwise regressive transfers of achievements from person to person in their common set of deprived dimensions, where the size of the transfer is nonnegative for any dimension in the set, and for at least one deprived dimension, the transfer has a positive size. In part (b) of condition (ii), it is ensured that the size of the transfer in any dimension does not allow the recipient to be nondeprived in the dimension. Finally, condition (iii) of the definition requires that in all the periods other than , social matrices in both and are the same. Since for any , cannot exceed , it is confirmed that .

This bundle of regressive transfers increases inequality in the achievement distributions among the poor. The following may be regarded as the chronic poverty ditto of this property:

1. Chronic Transfer: For all , , , , if is obtained from by a Pigou–Dalton bundle of regressive transfers between two chronically poor persons, then .

In the matrix , persons 1 and 2 are chronically poor, and person 2 has higher deprivations than person 1 in both the dimensions in period 1. Then a Pigou–Dalton bundle of regressive transfers consisting of sizes of 1 and 0.5 units of achievements in dimensions 1 and 2, respectively, from person 2 to person 1 will raise chronic poverty.

1. Chronic Dimensional Monotonicity: For all , , , suppose that person is chronically poor in , and is related to as follows: , for a triplet , and for all triplets , where . Then .

This property insists that chronic poverty goes up when a chronically poor person who is nondeprived in a dimension in some period becomes deprived in the dimension in that period. In the distribution matrix , person 1 is chronically poor but nondeprived in dimension 2 in period 4. If this achievement falls down to a level below the threshold limit, say from 5 to 3, then the person becomes deprived in dimension 2 in period 4. This expansion in the number of deprived dimensions of the chronically poor, person 1, should augment chronic poverty.

The next axiom is concerning association between dimensions of well-being. Consequently, it represents a unique feature of multidimensional chronic poverty analysis.

1. Definition: Suppose that persons and are chronically poor in the intertemporal social matrix . Assume also that they are deprived in dimensions and in some period and (i) , (ii) , (iii) for all . Next, suppose that is obtained from as follows: (iv) , (v) , (vi) for all , and (vii) for all , where is the lth row of . If for all , then we say that is obtained from by a correlation-increasing switch between two chronically poor persons. Conversely, it can be said that is derived from by a correlation-decreasing switch between two persons who are chronically poor.

Conditions (i) and (ii) of the aforementioned definition state that in , person has lower achievement than person in dimension , and the reverse inequality holds in dimension . According to condition (iii), in all other dimensions, achievements of person are at least as large as those of person . Condition (iv) formally states that in period , a switch between achievements of persons and in dimension has been performed. Conditions (iv) and (v) jointly ensure that in the postswitch situation, person has higher achievements than person in both the dimensions and in the period. This swap of achievements between persons and increases the correlation between the dimensional achievements. Note that in in no dimension, person has higher achievement than person .

1. Increasing Chronic Poverty under Correlation-Increasing Switch: For all , , , , if is obtained from by a correlation-increasing switch between two chronically poor persons, then given that the two dimensions affected by the switch are substitutes.

Similarly, when the underlying dimensions are complements, the reverse inequality holds. If the dimensions are independents, the perfect equality is achieved.

Suppose that the matrix is derived from by a switch of achievements in dimension 2 of persons 1 and 2 in period 1. Assume that all other entries in the two matrices are the same. We can then say that is obtained from by a correlation-decreasing switch.

1. Chronic Monotonicity in Threshold Limits: For all , , , suppose that person , who is chronically poor in , is deprived in dimension in period so that . Then , where , for , for , and is constant.

This postulate asserts that chronic poverty increases under an increase in the threshold limit of a dimension in which a chronically poor person is deprived in some period. In the social matrix , person 1 is chronically poor and deprived in dimension 2 in period 1. If the threshold limit of the dimension increases from 4 to 5, say, then the person's deprivation in the dimension in period 2 also goes up. This increased deprivation of person 1 leads to a higher chronic poverty. More precisely, .

The following axioms also seem appropriate for a duration-based index.

1. Chronic Boundedness: For all , , , , where (a) if , and (b) if for all triplets .

This axiom claims that the chronic poverty index is bounded between 0 and 1, where the lower bound, showing minimal poverty (0), is achieved if there is no chronically poor person in the society. In contrast, the index attains its upper bound, representing maximum poverty (1), when everybody in the society is maximally deprived in all the dimensions in all periods.

The next axiom is self-explanatory.

1. Chronic Continuity: Given , for any arbitrary population size , varies continuously with changes in achievements in each period provided that individuals' poverty statuses remain unchanged in all the periods.

5.5.2 The Bourguignon–Chakravarty Approach to Chronic Multidimensional Poverty Measurement

This subsection proposes a scalar representation of multidimensional chronic poverty in a society and evaluates it with respect to the axioms introduced earlier. It follows the Bourguignon and Chakravarty (2003) aggregation rule and, in consequence, can be regarded as the chronic twin of the Bourguignon–Chakravarty multidimensional poverty index.

The numerical representation of multidimensional chronic poverty, we suggest, is given by

5.33

where , are sensitivity parameters along the Bourguignon–Chakravarty lines of multidimensional poverty measurement; with , , and being arbitrary. As we will see, adoption of this functional form will capacitate us to identify any pair of dimensions in terms of a well-defined notion of association between them. This chronic poverty metric is time replication invariant as well.

Person i's deprivation indicator in dimension in the tth period is given by . The third bracketed term in (5.33) is the simple unweighted average of transformed values of such indicators over all dimensions in the period for the chronically poor person , where the transformed values are generated by applying the nonnegative increasing, strictly convex transformation , . These periodic dimensional averages are then aggregated across periods to arrive at , where . These quantities have been calculated for any arbitrary chronically poor person in the population. If we sum these values over all chronically poor persons in the society and divide the resulting expression by the population size , then we arrive at the summary standard in (5.33).

Now, person is fully deprived in any dimension in a year , if . If this situation of maximum deprivation occurs for all periods in all the dimensions for a chronically poor person , then the sum becomes for all and . If all the persons in the society are chronically poor and are characterized by maximum deprivation for all (period, dimension) combinations, then the value of the expression is . This in turn establishes that is bounded above by 1. It attains the lower bound 0 if nobody is chronically poor.

The following proposition, which is easy to demonstrate, describes some properties of .

The index can be expressed as the sum of dimensionwise indices when . Using this breakdown, we can calculate contribution of each dimension to the chronic poverty of the society (see Chapter 3). One can search for the factors that are likely to relegate people into poverty. Given a positive , as , approaches , where is the chronic head-count ratio, the proportion of persons that are chronically poor in , and is the average number of (period, dimension) combinations in which the chronically poor persons are deprived, that is, , expressed as a fraction of dT. The normalized quantity is the average duration of poverty among all chronic multidimensionally poor persons. Then the fraction is the multidimensional twin of the duration-adjusted head-count ratio. On the other hand, for , the index becomes , the sum of deprivations experienced by the chronically poor persons across all dimensions and periods, divided by , the maximum possible value that this number can assume. We can write it alternatively as

5.34

The component of this multiplicative decomposition is the average deprivation of the chronically poor persons in the poverty spells across all dimensions and periods. Each of the two terms of this simple multiplicative decomposition conveys significant information on different aspects of chronic poverty, confronted by the society. The first term tells us the extent of poverty arising from the fraction of population affected by chronic poverty. When this term divides the overall chronic poverty value , we are left with the third bracketed term of the decomposition, which indicates the average depth of deprivation felt by the chronically poor persons in all dimensions throughout the period profile . Evidently, while the former simply counts the proportion of persons affected by chronic multidimensional poverty, the latter represents its intensity. Each of these factors is quite important from policy perspective.

In order to illustrate the calculation of , let us consider again the social matrix . Assume, as before, that and . Recall that under these specifications, only persons 1 and 2 are chronically poor. Then the corresponding dimensional deprivation matrix whose entries represent deprivations of the chronically poor persons is given by

The entries in the 1st, 3rd, 5th, and 7th rows of indicate deprivations of person 1 in the two dimensions in periods 1, 2, 3, and 4, respectively. Entries in the 2nd, 4th, 6th, and 8th rows register similar figures for person 2. Since person 3 is chronically nonpoor, we do not include his deprivations in the matrix. The sum 3.4 of all possible entries in divided by is the value of when . This value then becomes . The head count ratio is . As a result, the average deprivation of the chronically poor person is . We can similarly calculate for other choices of and .

In our analysis in the earlier subsection, if we assume that the duration threshold , then the resulting situation should include deprivations of all the poor persons for the determination of overall poverty. For any , , and , we denote this by . In contrast, for arbitrary , , and , any restriction imposing a given positive value of means that takes into account only deprivations of the chronically poor persons. In consequence, it is reasonable to regard the difference

5.35

as a transient multidimensional poverty standard, where , , and are chosen arbitrarily. The decomposition (5.35) capacitates us to judge the contributions of both transient and chronic poverty, which may be helpful in analyzing the lifetime poverty.

Recall that in , person 3 is deprived only in the dimension 1 in period 4 and the value of the corresponding deprivation indicator is 0.4. This deprivation of person 3, in addition to those that are relevant to , should be included for the calculation of the overall poverty . For , the aggregate poverty is . Consequently, the level of the related transient poverty is . The head-count ratio also increases from to 1, which shows that the transient head-count ratio here is .