6.1 Introduction

Vulnerability is concerned with security risks. We can broadly define it in terms of a system's exposure and capacity to deal adequately with distress. For instance, a situation of economic vulnerability arises when a country faces an economic shock. Similarly, an ecosystem's exposure to climatic shocks may be regarded as a case of environmental vulnerability. A farmer with a low income from agriculture may be nonpoor currently. But since his agricultural output depends on the weather conditions, he may become poor in the future if the weather badly affects production. In the dimension of health, vulnerability may be regarded as a situation where a person with a reasonably good health condition currently will undergo an incident of health problem so that he becomes health-poor over time. (See Dercon and Krishnan, 2000, for an illustration). A person with a contractual nature of employment may be vulnerable to unemployment in the future (see, for example, Basu and Nolen, 2005).

From the aforementioned illustrations, it is clear that the notion of vulnerability is forward-looking. In the study of vulnerability, our concern should be not only with current conditions, such as income and health status, but also with the risks a person faces and his ability to avert, bring down, and conquer these. On the other hand, in the standard poverty analysis, both intertemporal and cross section, the analysis is based on observable information, more precisely, on the assumption of complete certainty. As a result, the analysis of vulnerability requires a separate treatment.

The study of vulnerability is quite important because of its highly significant implications for economic efficiency and long-term individual welfare. Many individuals face adversity in terms of continued illness, natural calamities, and other risks. These people can fall into poverty in the wake of adverse shocks. In consequence, the removal of vulnerability should be a concern of high priority from policy perspective. In the words of Sen (1999, p. 1): “The challenge of development includes not only the elimination of persistent and endemic deprivation, but also the removal of vulnerability to sudden and severe destitution.” Protection of “vulnerable groups during episodes of macroeconomic contraction is vital to poverty reductions in developing countries” (World Bank, 1997, p. 1).

Klasen and Povel (2013) argued that vulnerability at the individual level can be broadly classified into the following four broad categories: (i) vulnerability representing uninsured exposure to risk; (ii) vulnerability as a quantifier of low expected utility; (iii) vulnerability indicating expected poverty; and (iv) vulnerability to poverty. Discussions on the first three categories of vulnerability were made, among others, by Hoddinott and Quisumbing (2003); Ligon and Schechter (2003), and Gaiha and Imai (2009) (see also Hoogeveen et al., 2004). Vulnerability to poverty was introduced and analyzed by Calvo and Dercon (2013). (See Chakravarty et al., 2016; Fujii, 2016, for recent discussions.) In order to discuss these four divisions of vulnerability in greater detail, we assume that income is the only dimension of well-being.

Vulnerability indicating uninsured exposure to risk determines the extent to which income shocks yield changes in consumption (see Townsend, 1994; Amin et al., 2003; Skoufias and Quisumbing, 2005). The concerns of this concept of vulnerability are changes in the current magnitudes of consumption and not current sizes of consumption. This approach ignores a person's temperament about risks.

Vulnerability manifesting low expected utility identifies vulnerability with variability in a positive monotonic way in the sense that an increase in variability is regarded as a higher level of vulnerability. In the theory of statistical decision-making, there has been a long tradition of employing the variance as a measure of risk (Rothschild and Stiglitz, 1970). A sophisticated formulation of this was developed by Ligon and Schechter (2003). To get an insight of the underlying central idea, we consider two individuals, each of whom has the same expected consumption, not below some exogenously given norm, in some future period. It is also true that while for the first person there is a positive probability of destitution in the future, for the second person no such risk exists. It is quite likely that positive probability of adversity in the future period will relegate the first person to a situation of vulnerability. Since vulnerability is a forward-looking concept, it is evident that the two individuals should not be treated identically in terms of vulnerability. According to Ligon and Schechter (2003), vulnerability in a setting of this type can be interpreted as low expected utility.

The Ligon–Schechter index of vulnerability is given by the difference between the utility received from a threshold income, income poverty line, and the individual's expected utility obtained from income in a vulnerable situation. A higher (positive) difference between the two utility values indicates a greater level of vulnerability. This notion of vulnerability regards a person as nonvulnerable if his income is not below the poverty line (see also Glewwe and Hall, 1998; Dercon, 2002; Coudouel and Hentschel, 2000). Since the formulation depends directly on the von Neumann–Morgenstern utility function, an important characteristic of this approach is that it takes into account an individual's attitudes toward risks in an explicit manner. The harshness and possibility of disturbance on individual welfare are incorporated directly into the framework because of nonconstancy of the utility function and probabilistic formulation. Since all the individuals are assumed to possess the same utility function, levels of vulnerability are comparable across persons. Elbers and Gunning (2003) extended the Ligon–Schechter framework over an infinite time horizon. In Morduch (1994), vulnerability was expressed in terms of deviations from the permanent income poverty line.

Vulnerability as expected poverty deals with the risk of an individual's income falling below the income poverty line. This notion of vulnerability was introduced by Ravallion (1988) and advanced and discussed further by Holzmann and Jorgensen (1999). An analysis of this approach was developed in a more formalistic manner by Chaudhuri et al. (2002), which expresses the probability that an individual's income will fall below an exogenous income poverty line. (See also Christiaensen and Boisvert, 2000; Chaudhuri, 2003). However, it does not incorporate the awareness about risks. An individual's status in terms of vulnerability relies simply on some expected income. Hoddinott and Quisumbing (2003) made an attempt to address this shortcoming by considering vulnerability as expected poverty using the Foster et al. (1984) poverty index. When the negative of poverty is interpreted as utility, the Arrow (1965)–Pratt (1964) absolute risk aversion measure, an index of the extent to which a person is risk-averse, for the underlying utility function, increases as the value of the associated parameter increases. However, empirical findings did not support such a risk preference unambiguously (see Hoddinott and Quisumbing, 2003 and Binswanger, 1981). Additional empirical applications of this approach can be found in Suryahadi and Sumarto (2003); Kamanou and Morduch (2004); Christiaensen and Subbarao (2005), and Günther and Harttgen (2009). In Pritchett et al. (2000), vulnerability has been defined in terms of the probability of tumbling into poverty in three consecutive periods.

The concept of vulnerability to poverty was initiated by Calvo and Dercon (2013). They established an axiomatic characterization of an index of vulnerability as a weighted average of future state-contingent deprivations, where the weights are the probabilities of state-contingent returns in the future. “Ligon and Schechter's measure is the expected poverty gap, whereas Calvo and Dercon's measure is the expected Chakravarty index and Kamanou and Morduch (2004) employs the expected Foster–Greer–Thorbecke (FGT) index” (Dutta et al., 2011, p. 645). More generally, these indices are essentially expected poverty indices. In a different contribution, Calvo and Dercon (2009) showed how their index, “which in itself was based on the Chakravarty measure of poverty” (op.cit., p. 46), can be amended as a dynamic and forward-looking index of vulnerability. Dutta et al. (2011) developed an axiomatic characterization of an index of vulnerability that relies explicitly on the current and future incomes. Therefore, while in the Calvo–Dercon approach, deprivations depend on future incomes, the Dutta–Foster–Mishra framework allows us to look at relative changes under vulnerability.

In a recent contribution, López-Calva and Ortiz-Juarez (2014) suggested a view of the middle class that relies on vulnerability to poverty. They employed panel data for income to determine the level of comparable income corresponding to a low probability of falling into poverty. This in turn defines the lower bound of the middle class income. The countries they have considered in their analysis are Chile, Mexico, and Peru.

The examples provided at the beginning of this section clearly show that, similar to poverty, vulnerability is a multidimensional phenomenon (see Calvo, 2008). The objective of this chapter is to study vulnerability to poverty from a multidimensional perspective. This notion of vulnerability represents the strains laid down by the threat of multidimensional poverty. As a background material, we present a brief review of one-dimensional measurement of vulnerability to poverty in the next section. Section 6.3 deals with an axiomatic analysis of multidimensional vulnerability to poverty. Finally, Section 6.4 winds up the chapter. A brief analysis of the Calvo and Dercon (2009) amended index is also presented in this section.

6.2 A Review of One-Dimensional Measurement

Since the one-dimensional approach to vulnerability to poverty measurement is closely related to the multidimensional analysis of the issue, we begin this section with a rigorous discussion on the former. For expositional ease, we assume that income is the only dimension of human well-being. In addition, initially the analysis is carried out at the individual level.

The indices of vulnerability we scrutinize here are based on anticipated changes, that is, they are ex-ante measures in the sense that they incorporate future uncertainty with reference to income. Thus, income is regarded as an uncertain prospect. An absolutely necessary characteristic of these indices is that the underlying risks are downside risks, that is, in the future, there is possibility of downward trend of income. Individual vulnerability here is developed in terms of shortfall of income from the exogenously given income poverty line resulting from economic and other shocks.

Since income is considered as an uncertain prospect, there are different levels of returns on the prospect. These returns are state-contingent or state-dependent outcomes. A state of nature is a situation for the prospect that can arise in the future. Accordingly, by a state-contingent return we mean a return whenever a particular state materializes. In order to illustrate this, consider a farmer for whom there is a high impact of weather conditions (rainfall) on crop production. Relevant states of nature are the rainfall conditions, say: (i) drought, (ii) less than barely sufficient but not drought, (iii) barely sufficient, (iv) optimum, and (v) more than optimum (flood). There is a return associated with each state. These are state-contingent returns.

Assume that the society under consideration consists of n individuals and k states, where is arbitrary and is an integer. We denote the set of states by . In order to compare individual vulnerabilities across persons, we will assume throughout the section that the set of states remain the same for all individuals in the society. For individual i, the associated state-contingent returns are represented by a vector , that is, is the unique income that individual i receives if state j emerges; , . As a consequence, when production of a crop gets affected by variations in rainfall, different levels of rainfall describe the states, and the level of production, when a particular state comes into perceptible existence, is the state-contingent return.

Individual i assumes that the probability of appearance of state j is . We denote the vector of probabilities by . Evidently, for all , and , . Since the return is uniquely associated with state j, we can as well say that comes into existence with probability . For any given state j, these probabilities are likely to vary across persons. That is, for any two persons i and h, and for any state j, need not be the same as , where and . To illustrate this, we consider two farmers, one of whom has easy access to deep tube well for pumping out underground water if the rainfall is inadequate for crop production. However, for the other person, such a facility does not exist. Therefore, if the rainfall for crop production is not at the requisite level, while it is highly unlikely that the former person's crop production will be badly affected by drought, for the latter individual, this chance is quite high. In other words, the probability of appearance of “drought” is quite low for the first person, whereas for the second person, the probability is quite high. In general, such differences arise because the probabilities may be affected by some external factors, which cannot be ignored. Hence, these probabilities are subjective probabilities – they are based on personal feelings.

Denote the exogenously given threshold limit, the income poverty line, by , and assume, as in Chapter 3, that , where , that is, z takes on values in a finite nondegenerate positive interval of , the nonnegative part of the real line . We say that individual i is poor in state j, equivalently, state j is meager for i, if . Otherwise, that is, if , the person is nonpoor in the state, equivalently, the state is nonmeager for him. Person i is identified as vulnerable if there is at least one state j such that and . This can be regarded as the vulnerable analog to the contention of identification of a poor person arising in an income poverty situation. A one-dimensional vulnerable situation involving the state-contingent returns , the corresponding probability vector and the poverty line may be expressed more compactly as , say, where is the nonnegative orthant of the k-dimensional Euclidean space , and is nonnegative k-dimensional unit simplex, the set of all k-coordinated vectors whose coordinates are nonnegative real numbers that add up to 1. Let denote the set of all one-dimensional vulnerable situations for person i, that is, . The censored state-contingent return vector associated with is denoted by , where .

Since we focus on the downside risk, that is, given that each has a chance of being less than z, the possibility of future poverty occurrence is considered in the framework. In order to determine vulnerability at the individual level, it may be worthwhile to consider the shortfalls of the state-contingent returns from the poverty line in different states of nature. For any person i, these shortfalls, when expressed in relative or proportionate terms, are given by . By definition, each of them is homogeneous of degree 0, that is, invariant under equiproportionate changes in the corresponding state-contingent return and the threshold limit. We refer to them as vulnerability deprivation indicators, which, for a given threshold limit, are decreasing in state-level returns. They are positive if and only if state-dependent returns are below the threshold limit. The issue of considering a vulnerable person's deprivations parallels the idea of looking into the poverty shortfall of an income-poor person.

For any person i, the situations of the type are transformed into vulnerability levels experienced by person i using a function . In more explicit terms, . The function can be designated as the individual vulnerability index. The presence of superscript i in explicitly recognizes that determines the extent of poverty suffered by person i under probable realization of different states. For any , determines the extent of vulnerability that the person undergoes.

We now suggest the following axioms for the individual vulnerability index. Variants of some of these axioms were discussed by Calvo and Dercon (2013), Chakravarty and Chattopadhyay (2015), and Chakravarty et al. (2015).

1. State-Restricted Focus: For any , if for some , , then , where .
2. State-Restricted Monotonicity: For any , if for some , , , then , where .
3. Across-States Transfer: For any if for some , then , where for all and .
4. State-Restricted Monotonicity in Threshold Limit: For any , if for at least one state , and , then , where and .
5. Severity of Downside Risks: For any , if for some , , then , where , , and .
6. State-Restricted Boundedness: The index is bounded between 0 and 1, where the lower bound is achieved if for any , is such that for all . reaches its upper bound if for any , for all .
7. State-Restricted Replication Principle: For any , suppose that for each , the state-contingent return is replicated m times and the corresponding probability value is split equally among the replicated figures of , then , where is the mk-coordinated vector of replicated returns and is the mk-coordinated vector of split probabilities, being any integer.
8. Continuity in State-Contingent Returns: For given , varies continuously with respect to variations in returns such that the vulnerability status of the person remains unchanged.

One common feature of these state-restricted properties is that they can be termed as individualistic axioms in the sense that the underlying operations involve only one person's vulnerability. The focus axiom here demands that an increase in the return from a nondeprived state does not change the vulnerability index. Since vulnerability is concerned with downside risk, this axiom is quite plausible. If the low return from a state with a positive probability of being deprived reduces further, then the extent of vulnerability increases. The current version of the monotonicity axiom makes this sensible claim. To illustrate this axiom, suppose that person i foresees that because of economic shocks, his current income 620 may reduce to 610, 590, and 580, respectively, with probabilities , and . Assume that the threshold income . Then and . Suppose that the person revises his prediction under some apprehension that the lowest anticipated income 580 will actually be 570. Now, given that the probabilities and the other anticipated incomes remain unchanged, we have and the monotonicity axiom demands that .

According to the present form of the transfer axiom, vulnerability goes up under a transfer of return from a state with low positive return and high downside risk to another state possessing a higher return but not higher risk. The intuitive reasoning behind this postulate is that downside risk is not lower in state q than in state j, and the transfer decreases the lower return in q further and increases that in j, which already had higher return. Hence, vulnerability should increase. Undoubtedly, such a transfer may be regarded as an across-states regressive transfer. Suppose in the aforementioned example, is generated from by a regressive transfer of income from state 3 to state 2. Since state 3 has a lower return but higher probability than state 2, according to the across-states transfer axiom, . We note the similarity and dissimilarity between the income poverty transfer axiom and the transfer axiom proposed here. While for a notion of transfer to be valid in the income poverty case, there should be at least two persons; here we need existence of at least two states for the prospect, and it concerns only one person.

If in a deprived state, which has a positive probability of appearance, the person becomes more deprived resulting from an increase in the threshold limit, then vulnerability should increase unambiguously. The variant of the axiom of monotonicity in poverty line suggested in the current section asserts this. Since vulnerability is concerned with downside risk, if this type of risk of return from a state with low return goes up and the corresponding risk from a state with high return goes down, then evidently vulnerability should demonstrate an upward trend. This legitimate requirement is affirmed by the axiom of severity of downside risk. This postulate is a unique characteristic of the vulnerability index and does not parallel any standard poverty property. Given that and , suppose that person i increases his subjective probability of falling his current income from 620 to 580 by and reduces that of falling to by . Hence, changes to . Of states 2 and 3, originally the latter had a lower return accompanied by a higher probability. Hence, the conditions laid down in the severity of downside risk axiom are satisfied. As a result, .

The boundedness axiom related to the one-dimensional vulnerability of a person insists that vulnerability index takes on a finite value lying between 0 and 1, where the minimum value 0 is attained if none of the returns falls below the threshold limit so that the person has a nonvulnerable status. It obtains its upper bound 1 if the return from each state is 0 so that the person is maximally deprived. Since there is no possibility of further reduction of the return from any state, this maximum value is achieved irrespective of the probability distribution. To understand the state-restricted replication principle, let and . Assume that each state-contingent return is replicated twice. Then and . The individual vulnerability levels of the two situations are the same. The final axiom, continuity in state-specific returns ensures smooth behavior of the vulnerability index with respect to variations in returns under ceteris paribus conditions. Note that change in one probability will require a change in at least one different probability, so the sum of all probabilities equals 1.

As an illustrative example, we consider the Calvo and Dercon (2013) individual vulnerability index defined as

6.1

where the constant controls risk awareness. As Dutta et al. (2011) noted, this is the expected Chakravarty (1983) poverty index. The restriction ensures that the focused, normalized, and continuous index satisfies the monotonicity (in returns), transfer, and downside risk severity axioms. It increases if the poverty threshold increases under the minor restriction considered. For , it is the proportionate gap between the poverty line and the expected state-dependent return. This is the situation of risk neutrality. On the other hand, as , . This index is a member of a family of expected poverty or deprivation indices given by , where satisfying is a transformed deprivation indicator. It is also assumed to be decreasing, continuous, and strictly convex in state-contingent return. Decreasingness and strict convexity ensure satisfaction of the monotonicity and the transfer axioms, respectively. The normalization condition guarantees attainability of the lower bound of . Depending on the forms of chosen, we have different vulnerability indicators. In the Calvo–Dercon case, .

If the transformed deprivation indicator is given by , where is a parameter, then the expected poverty index turns out to be the expected Foster et al. (1984) poverty index (see Kamanou and Morduch, 2002):

6.2

Under the parametric restriction , all the vulnerability axioms are abided by . For , coincides with if . For a given , as the value of increases, more weight is assigned to higher proportionate gaps. In addition, given , an increase in the value of does not increase the value of the index.

Assuming that all the state-contingent returns are positive, we may use as a transformed deprivation indicator. The resulting vulnerability index becomes the expected Watts (1968) individual poverty function, defined as

6.3

Because of positivity restriction of the state-contingent returns, is not bounded above. However, it meets all other axioms.

As the final illustrative example, we consider the Bourguignon and Chakravarty (2003) individual vulnerability index defined as

6.4

where and are parameters. This index agrees with all the vulnerability postulates we have analyzed. It, however, does not fit in the structure unless , in which case it coincides with .

We define the one-dimensional global or overall vulnerability index as a nonnegative real-valued function of all one-dimensional vulnerable situations of different persons. As we have assumed, the set of states is the same for all persons, and the threshold income is exogenous. For a set of population consisting of n individuals, where is arbitrary , let denote the set of all one-dimensional vulnerable situations, when all the individuals are taken together, that is, . Let , or, (for short) stand for any arbitrarily chosen . By definition, . The set is the domain of our global vulnerability index . Formally, .

As an illustrative example, we regard as the arithmetic average of individual vulnerability indices (see also Calvo and Dercon, 2013). Then our global vulnerability index is defined as

6.5

where is arbitrary. The idea of defining the aggregate vulnerability index as a function of individual indices may be regarded as representing a property, which we can refer to as independence of irrelevant information. This is because all individualistic nonvulnerability features are ignored in this definition.

By construction, the vulnerability index in (6.5) obeys subgroup decomposability. Thus, it has nice policy relevance in the sense that it becomes helpful in identifying higher vulnerability affected subgroups that may become a policy adviser's targeted subgroups for reducing social vulnerability.

It possesses several interesting properties, assuming that unambiguously meets all the axioms proposed earlier.

1. i. It is bounded between 0 and 1, where the lower bound is achieved if everybody in the society has a nonvulnerable status. The upper bound is attained if all the individuals are subject to maximum deprivation in all states. (For the index associated with (6.3), the upper bound is not attained.)
2. ii. It varies continuously for changes in meager achievement levels, under certain ceteris paribus conditions.
3. iii. It meets a strong monotonicity property in the sense that an increase in an individual's income vulnerability increases its value. This property parallels the strong Pareto principle analyzed in Chapter 1.
4. iv. Discussion on the next postulate relies on the following definition.

In state q, person i has higher deprivation than person h in state j, and the corresponding probability for i's deprivation is not lower than that for h's. The regressive transfer (of size ) takes place from person i's return in state q to the return for person h in state j. Since the probability vectors in the two situations and the threshold limit remain the same, the regressive transfer should increase global vulnerability. Further, given that the regressive transfer occurs between two persons, we can refer to it as a one-dimensional nonindividualistic transfer. In addition, it involves two states. We can present the following axiom for our one-dimensional overall vulnerability index .

1. Across-States One-Dimensional Nonindividualistic Regressive Transfer: If the one-dimensional global vulnerable situation is obtained from the situation by a nonindividualistic regressive transfer, .

1. i. Any reordering of individual vulnerability levels does not change the value of . Hence, the individuals are separated only by vulnerability levels.
2. ii. Any m-fold replication of the population leaves global vulnerability unchanged, where is an integer. In consequence, this characteristic enables us to compare vulnerabilities of two societies with different population sizes. This population replication principle is stated under the supposition that the vectors of state-contingent returns and probabilities remain unchanged.

Although properties stated in (i)–(vi) are examined for the global index that meets subgroup decomposability, they may as well be regarded as desirable postulates for a general global vulnerability index , which may not be subgroup decomposable. One such possible example is , where .

We conclude this section by providing one illustration of using as the individual index. The resulting global index turns out to be

6.6

where, as before, and are parameters. The major difference between this index and the overall Foster–Greer–Thorbecke index, which arises when , and the Calvo–Dercon expected Chakravarty index is that, while in the former the individual expected income poverty levels are aggregated nonlinearly across persons, the latter two employ a simple linear aggregation.

6.3 Multidimensional Representation of Vulnerability to Poverty: An Axiomatic Investigation

As we have argued in several chapters earlier, the well-being of a population is a multidimensional issue. Hence, to get a complete picture of individual and social vulnerabilities in a population, it is necessary to investigate the problem from a multidimensional perspective. Since in the capability-functioning approach, capability failure captures the notion of deprivation that people experience in day-to-day living conditions, it is a multidimensional phenomenon. Consequently, multidimensional vulnerability to poverty is a problem of expected capability deprivation.

As in Chapter 3, we denote the number of dimensions of well-being in a society consisting of n individuals by d, where is arbitrary. We denote the set of all dimensions by Q. The d-dimensional row vector of poverty thresholds or threshold limits in different dimensions is the vector . The notion of vulnerability we consider, in this section, is a multidimensional distress in the sense that the affected person faces descending trend of dimensional achievements. More precisely, there are downside risks that the individual's achievements in different dimensions drop down to levels lower than the corresponding thresholds.

For any dimension j, there are states of nature, and the set of such states is designated by , . Since different dimensions reflect different aspects of well-being, it is unlikely that the types and numbers of states will be the same across dimensions. Let and stand, respectively, for person i's achievement in state s of dimension j and the probability of appearance of ; , . Evidently, and for all and .

The dimensional row vector of returns for person i from dimension j and the corresponding probability vector are given, respectively, by and . For any , each triplet is an element of , where is that subset of the positive part of the real line in which can vary. More precisely, for any , , where and are arbitrary. Given that there are d dimensions of well-being, for any , a multidimensional vulnerable situation is of the form . Let stand for the set of all multidimensional vulnerable situations for person i, that is, .

We follow the union method of identification in this multidimensional framework, that is, person i is called multidimensionally vulnerable if there is at least one dimension j and one state such that and . Let be the vector of censored returns for person i from different states in dimension j, where , with and being arbitrary. We similarly symbolize by .

The extent of destitution felt by person i in state is measured by the deprivation indicator , which we designate by , for short; , . The vector stands for the vector of such indicators associated with . For any person , the expected poverty in dimension j corresponding to is , where the transformed deprivation indicator satisfying is continuous, decreasing, and strictly convex in state-contingent returns. For any and , is based on the returns censored at .

The task of a multidimensional vulnerability index for person i is to summarize the information contained in the multidimensional vulnerable situation of the type for the person in terms of a nonnegative real number. Hence, the domain of the nonnegative real-valued function is . More precisely, .

Most of the individualistic axioms suggested in the earlier section, for a one-dimensional individual vulnerability index, can be adapted in the multidimensional system under certain alterations in the statements. For the sake of completeness, next we state these axioms analytically.

1. State-Restricted Weak Focus: For any , if for all , , , then , where , , for all ; for all , ; and for any , .
2. State-Restricted Strong Focus: For any , if for some , , , then , where , , for all ; for all , ; and for any , .
3. State-Restricted Monotonicity: For any if for some , , , , then , where , , for all ; for all , ; and for any , .
4. Across-States Transfer: For any if for some ; , , then , where , , , where ; for all , ; and for any , .
5. State-Restricted Monotonicity in Threshold Limits: For any , if for at least one , and , then , where, , , for all and .
6. Severity of Downside Risks: For any if for some , , then , where , , ; for all ; for all , ; and for any , .
7. State-Restricted Boundedness: The index is bounded between 0 and 1, where the lower bound is achieved if for , for all , and . reaches its upper bound if for , for all , and , assuming that this case of maximal deprivation is well defined.
8. Continuity in State-Contingent Returns: For given probability vectors and threshold limits, varies continuously in state-contingent returns, assuming that the vulnerability status of the person remains unchanged.

For a given profile of probability vectors and vector of threshold limits, if a person is not deprived in any state of any dimension, then giving him more return in some state of some arbitrary dimension should not have any impact on his vulnerability. The state-restricted weak focus axiom demands this. The remaining axioms stated in the section are simple multivariate translations, involving states of nature in different dimensions, of univariate vulnerability axioms specified in Section 6.2. Our discussion on postulates for the one-dimensional indices applies equally well here under obvious modifications.

An example of an individual vulnerability index in this multidimensional framework can be

6.7

where is arbitrary and . Here the expression can be regarded as expected poverty from dimension j ascertained by the Bourguignon–Chakravarty multidimensional vulnerability index for person i. The parametric restrictions and are sufficient to guarantee that agrees with all the axioms laid down for an arbitrary .

For defining an aggregate multidimensional vulnerability index for a given set of population with n individuals, we denote the arbitrary vulnerable situation by , or, by (for short), where and . Further, let stand for a multidimensional global vulnerable situation . The set of all multidimensional global vulnerable situations is . Let stand for the d dimensional vector corresponding to , where is person i's expected poverty in dimension j.

The arithmetic average of individual multidimensional vulnerability indices can now be taken as an index of the aggregate multidimensional vulnerability index. Accordingly, for any arbitrary population size of , this global index is defined as

6.8

where is arbitrary for each i and fulfills the axioms stated in the section.

A clear distinction exists between the aggregation rule employed in (6.8) and the dashboard and composite index approaches. As we have observed, different dimensional indices of deprivation (or well-being) can be assembled in a set, framing the dashboard approach. On the other hand, when the dimensional indices are united, using some aggregator, we have a composite index. It may be recalled here that one common problem with dashboards and composite indices is that they ignore joint distributions of deprivations or well-beings.

Observe that in (6.7) statewise deprivations of person i in a dimension are aggregated initially. The aggregated dimension level figures are clubbed together to arrive at . Then in (6.8), we have a vector of individual multidimensional indices that are combined using the unweighted arithmetic averaging criterion. By construction, joint distribution of deprivations across the dimensions is explicitly taken into account.

In view of satisfaction of population subgroup decomposability by , its policy applicability for pinpointing population subgroups that are distressed more by multidimensional vulnerability is evident. Several attractive features of this aggregate vulnerability index are analyzed next.

1. i. It reaches the lower bound 0 when everybody is nondeprived in all states of different dimensions. In contrast, it arrives at the upper bound 1 if all the persons are maximally deprived in all the dimensions.
2. ii. It is continuous in achievement quantities under certain mild conditions.
3. iii. Because of increasingness in individual arguments , where , it is strongly monotonic. Therefore, any change in an individual index, as desired by the axioms, specified earlier, is taken into account by the index properly.
4. iv. It also fulfills a multidimensional regressive transfer principle, which we define next.

In the aforementioned definition, given that all the probability distributions and the threshold limits are the same across the profiles and , we can say that is obtained from by a regressive transfer of achievement in state from person h to person i. In , person h, the donor of the transfer, has higher deprivation than person i, the transfer recipient, in state s of dimension j, and the corresponding positive downside risk is also not lower for the donor. This regressive transfer of achievement in state s of dimension j makes the donor more deprived in the state, where , and are arbitrary. Both the numbers of dimensions and the corresponding states for which (i) and (ii) hold can be more than 1. Accordingly, it is a multidimensional phenomenon. Given that the donor has at least a high positive downside risk than the recipient in the state under consideration, the regressive transfer that generates from should increase vulnerability.

To get more insights into this definition, let , , , and . Assume that , , , ; , , , and . In state 3 of dimension 2, person 2 has a higher deprivation but lower probability than person 1. Therefore, a regressive transfer between the two persons in state 3 of dimension 2 is ruled out. Since in state 1 of the dimension, the two persons are equally deprived, a regressive transfer is not possible here as well. Finally, in state 2 of the dimension, person 1 is not deprived but person 2 is deprived. In consequence, this state does not come under the purview of a transfer. Continuing this way, we can identify that only in state 3 of dimension 1, a regressive transfer from person 2 to person 1 is permissible.

The following axiom can now be stated formally:

1. Multidimensional Vulnerability Regressive Transfer: If the multidimensional global vulnerable situation is obtained from the situation by a multidimensional regressive transfer, .

1. v. Given that the index is simple unweighted average of individual multidimensional vulnerability indices, it is a symmetric function of individual indices. In other words, this postulate states that any permutation of individual vulnerability extents keeps its value unchanged.
2. vi. Since the index is defined as the average of individual multidimensional vulnerability indices, it remains unaltered under replications of the population.

Even though properties scrutinized in (i)–(vi) are specified for the subgroup decomposable index , they can be taken as intuitively reasonable postulates for a general multidimensional vulnerability index , which may not be subgroup decomposable.

As an example of a multidimensional vulnerability index at the society level, we may suggest the use of the following:

6.9

where for any , ; and . Equation (6.9) can be rewritten in terms of individual expected poverties for different dimensions as

6.10

The subgroup decomposable Bourguignon–Chakravarty multidimensional vulnerability to poverty index , encompassing the whole population, is bounded between 0 and 1, strongly monotonic, symmetric, population replication invariant, and correctly responsive to the multidimensional vulnerability regressive transfer principle for all and . (See properties stated in (i)–(vi) earlier.)

6.4 Concluding Remarks

In this chapter, we have only addressed the problem of representing a vulnerable situation, formulated in a particular way, numerically. In a recent paper, Chakravarty et al. (2015) developed a vulnerability ordering that regards one situation (e.g., agriculture) as not less vulnerable than another (say, fisheries) if and only if the former does not have lower level of vulnerability than the latter for a family of expected poverty indices, assuming that income is the only dimension of well-being. The family includes expected poverty indices, where the underlying transformed deprivation indicators are continuous, normalized, and convex. While such indicators we have discussed in Section 6.2 are homogeneous of degree 0 in the state-level returns and the threshold limit, the ordering does not require this property. Expected poverty indices with translation invariant transformed deprivation indicators that remain unchanged under equal absolute changes in such returns and the threshold limit can as well be included in this family. An example of such a function is the Zheng (2000) transformed deprivation function defined by , where is a constant. This function is decreasing, continuous, and strictly convex in state-dependent returns below the threshold limit. The main result developed by Chakravarty et al. (2015) does not require equality of the number of states across the situations. It banks explicitly on Blackwell's (1951, 1953) well-known results for comparisons of experiments (see also Cremer, 1982 and Leshno and Spector, 1992). A novelty of this ordering is that it can be regarded as vulnerability analog to the Hardy et al. (1934) classical result on the measurement of inequality. Hardeweg et al. (2013) considered a stochastic dominance-based partial ordering in this context. The multidimensional extension of these orderings is an issue of a natural investigation here. We have also not incorporated ordinally measurable dimensions into our analysis. Research on axiomatic approach to vulnerability has just begun. Many more relevant aspects are yet to be explored.

In a recent contribution, Chakravarty et al. (2016) investigated the implications of vulnerability on the income poverty line. More accurately, the problem of adapting the income poverty threshold under vulnerability so that the adjusted poverty line also represents the subsistence standard of living in a situation of vulnerability has been addressed. The central idea underlying this process of modification is that the individual utility derived from the existing poverty line and the expected utility generated by the new poverty line affected by a random error (noise) indicating vulnerability are the same. Consequently, the formulation relies on the implicit assumption that the vulnerability is treated as a low expected utility situation. Under certain sensible assumptions about the noise, in an additive model, the harmonized poverty line is shown to exceed the existing poverty line by a constant amount if the utility function possesses constant Arrow–Pratt absolute risk aversion. Likewise, in a multiplicative model, the adjusted poverty line becomes a scale transformation of the existing poverty line, where the underlying scalar is greater than unity, if the utility function exhibits constant Arrow–Pratt relative risk aversion. (The relative risk aversion measure is obtained by multiplying its absolute counterpart by income.) An empirical illustration of the developed methodology has been provided using data from the Asia-Pacific region. Clearly, an extension of this approach to the multidimensional setup will require consideration of joint distribution of the noise term representing vulnerability.

In an earlier contribution, Dang and Lanjouw (2014) developed two formal approaches to the determination of the vulnerability line. According to the first approach, for a population subgroup that is clearly not vulnerable, the vulnerability line has been defined as the lower-bound income of subgroup. In contrast, in the second approach, a subgroup that is not poor currently but faces a real risk of falling into poverty is considered. The upper-bound income for this subgroup has been taken as the vulnerability line. While essential to the Chakravarty et al. (2016) approach is the Arrow–Pratt theory of risk aversion, the Dang–Lanjouw approach relies on a probabilistic formulation.

In Chapter 5, we have assumed perfect foresight of period-by-period achievements on different dimensions for all the individuals. In their highly interesting contribution, Calvo and Dercon (2009) suggested one-dimensional statistic of vulnerability in a dynamic world in which the assumption of realization of incomes in the future periods with certainty is relaxed. Let stand for person i's income when state materializes in period t, where denotes the set of states of nature in period and . We write for the corresponding censored income. Let be the probability that state in period t emerges. Evidently, for all and , . The expected value of transformed normalized (censored) incomes in period t is , where is a constant (see Eq. (6.1)), E denotes the expected value operator, and z is the income poverty line.

The authors proposed the use of the following modified version of (5.12)

6.11

as a forward-looking and dynamic metric of poverty, where is the profile of the vectors of person i's incomes associated with different sates of nature in all T periods, is the corresponding profile of probability vectors, and is the rate of time discounting. This index can be regarded as a representation of upcoming intertemporal individual poverty because it is not based on evaluation of poverty for a single period; instead, it takes into account poverty assessments for a sequence of forthcoming periods in a world of uncertainty. According to the authors, “one of the contributions of this paper is to identify the Chakravarty poverty index as the best choice if the poverty analysis moves from static poverty on to vulnerability” (Calvo and Dercon, 2009, p. 57). Clearly, this index can be employed for arriving at a complete ordering of different possible paths of standard of living, judged by utilizing future uncertain incomes. A natural generalization of this nice proposal with many innovative features is to develop analogous quantifiers in the multidimensional setup.

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