Measuring Economic Insecurity∗

Walter Bossert Department of Economics and CIREQ, University of Montreal [email protected]

Conchita D’Ambrosio Universit`a di Milano-Bicocca, DIW Berlin and Econpubblica, Universit`a Bocconi [email protected]

This version: December 27, 2009

Abstract. We provide a systematic treatment of the measurement of economic insecurity, assuming that individual insecurity depends on the current wealth level and its variations experienced in the past. Current wealth could also be interpreted as incorporating the individual’s evaluation of future prospects. Variations in wealth experienced in the recent past are given higher weight than experiences that occurred in the more distant past. Two classes of measures, related to the generalized Gini social evaluation functions, are characterized with sets of plausible and intuitive axioms. Journal of Economic Literature Classification No.: D63. Keywords: Insecurity, Wealth Streams, Social Index Numbers.

∗

We thank Miguel A. Ballester, Lars Osberg, Erik Thorbecke, John A. Weymark and

participants at several conferences for helpful suggestions and comments. Financial support from MIUR (Prin 2007), the Fonds de Recherche sur la Soci´et´e et la Culture of Qu´ebec and the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.

1

Introduction

There is a perception that the level of economic insecurity has risen recently, in particular in view of the 2008 global economic crisis. This is by no means a new phenomenon— economic agents have struggled with sentiments of insecurity frequently in the past. In spite of the persistence of this occurrence, it appears that a formal definition and precise suggestions for how to measure it have remained elusive so far. Osberg [20] observes that the term economic insecurity did not appear in dictionaries of economic jargon at the time. Although more than a decade has passed since Osberg’s [20] observations, there does not seem to have been much progress in the interim in this regard—standard reference works such as the most recent editions of the New Palgrave Dictionary of Economics, the International Encyclopedia of Social and Behavioural Sciences and the Social Science Encyclopedia, to name but a few, still do not include an entry on economic insecurity. In addition, to the best of our knowledge, there are very few papers in the economics literature which have dealt with defining and measuring it. Osberg [20] notes that a considerable amount of research has been done on economic risk but, in contrast, insecurity has not received much attention. He suggests that a plausible definition of insecurity, consistent with the common usage of the term, might be based on a sentiment of anxiety generated by the lack of economic safety. In other words, insecurity may arise from the possible exposure of an economic agent to adverse occurrences, combined with the agent’s perceived difficulties to deal with their negative consequences. In order to arrive at a more precise definition, we first note that an intertemporal framework appears to be called for: past, present and future are all involved in the above intuitive description. We are insecure about the future, since the future is all that matters for generating anxiety to an individual—we know where we are today and the past cannot be changed. The resources we have today are important: the wealthier we are, the bigger the buffer stock we can rely on. Finally, our experiences play a crucial role in shaping our perception regarding how well we can manage in case of an adverse event. Clearly, we remember gains and losses in our resources that we experienced over time. The more recent these variations are, the more vivid are our memories; evidence from sources in psychology and economics support this view. People frequently decide from experience as opposed to deciding from description, that is, people’s choices are based on previous personal experience and not on the description of all possible outcomes and their probabilities. For a comparison among the two concepts, see, among others, Hertwig, Barron, Weber and Erev [15]. Recent studies have highlighted that personal experience

1

determines investors’ behaviour. Kaustia and Kn¨ upfer [16] found a strong positive link between past initial public offerings returns and future subscriptions at the investor level in Finland as predicted by reinforcement learning theory. Similarly, Malmendier and Nagel [18] have shown that in the USA individuals’ experiences of macroeconomic outcomes have long-term effects on their risk attitudes and willingness to bear financial risks. Personal experience seems also to have a wider influence on the attitudes of people. Fern´andez, Fogli and Olivetti [13], for example, have argued that the growing presence of men brought up in a family in which the mother worked is a significant factor in the increase in female labor force participation over time. We need to identify the economic variables that influence individual perceptions of insecurity. While there are, of course, many aspects of life that may play an important role, it seems to us that an adequate (and, from an applied perspective, realistic) option is to use a comprehensive notion of wealth as the relevant variable. We postulate that an individual’s sentiment of insecurity today depends on the current wealth level and variations in wealth experienced in the past. Wealth is assumed to encompasses everything that may help an individual in coping with adverse occurrences. It includes, for instance, claims on governments, family, friends etc. Sen [25] refers to these claims as entitlements, that is, “the set of alternative commodity bundles that a person can command in a society using the totality of rights and opportunities that he or she faces” (Sen [26, p.497]). It is important to note that the current wealth level could also be interpreted as the present value of all expected changes in future wealth. In this case, this value will take into account an individual’s evaluation of future prospects. Because we do not want to commit to any specific theory of choice under uncertainty, we leave the way that expectations about the future are integrated into current wealth open and focus on the aggregation problem that remains: the way we might want to solve trade-offs involving wealth variations in the past and current (including expectations) wealth levels. Thus, the measures of individual insecurity we propose in this paper have as their domain wealth streams of varying lengths. The length of these streams is not assumed to be fixed due to the observation that individuals are of different ages at a given time period and, moreover, the availability of data may impose restrictions on how far back in the past we can go when assessing economic insecurity. We propose a set of three basic properties that we think a measure of economic insecurity should possess. We then move on to more specific classes of measures that, in spirit, are related to the generalized Gini social evaluation functions. According to these measures, insecurity is given by the current wealth level multiplied by minus one plus 2

weighted sums of the wealth gains (losses) experienced in the past. Two sequences of coefficients are employed—one applies to gains, the other to losses. The coefficients are such that recent experiences are given higher weight than experiences that have occurred in the more distant past. A subclass of these measures is obtained by giving higher weights to the absolute values of past losses than to those of past gains, reflecting an attitude analogous to risk aversion in models of individual decision making under uncertainty. Both the class of two-sequences Ginis and the latter rank-ordered subclass are characterized with sets of plausible and intuitive axioms. The remainder of the paper is organized as follows. The next section presents the formal framework and the properties we think a measure of economic insecurity should possess. In section 3 we characterize the class of two-sequences Gini measures. The rankordered subclass of these measures is characterized in section 4. Section 5 concludes with a discussion of how our indices distinguish themselves from criteria known from inequality measurement and the theory of risk.

2

Wealth streams and individual insecurity

For any T ∈ N0 , let R(T ) be the (T + 1)-dimensional Euclidean space with components labeled (−T, . . . , 0). Zero is interpreted as the current period and T is the number of past periods taken into consideration (which may vary). A measure of individual insecurity is a

 sequence of functions V = V T T ∈N0 where, for each T ∈ N0 , V T : R(T ) → R. The measure assigns a degree of insecurity to each individual (net) wealth stream w = (w−T , . . . , w0 ) ∈ S (T ) . We allow wealth to be negative. The wealth stream w = (1, 3, 3, −1, 0, 2) ∈ T ∈N0 R (5) R is illustrated in figure 1.

3

wt

3

2

1

0

•

•

•

−5 −1

•

............................................................ ... ... ... ... ... .. . ... .. . ... .. . ... .. . ... .. ... . . . ... . ... ... . ... .. . . ... .. . ... ... .. .. . ... .. .. . . . ... . . ... ... ... ... ... ... ... ... ... ... .. .. . . ... .. ... ... .. ... ... ... ... ... ... .. .. ... . ... ... ... ... ... .. ... .. . . ... ... ... ... ... ... ... ... ... . ... ... ... .. ... ... ... .... . . . . ... . ... .... ... ... .... ... ... ... .... . . . ... .... ... ... ... ... ... ...... ........ ...

−4

−3

−2

• −1

t

0

•

Figure 1: The wealth stream w = (1, 3, 3, −1, 0, 2). It is possible to think of w0 as encompassing not only the current wealth level of the individual but also her or his assessment of (uncertain) future levels of net wealth. We do not want to commit to a specific method of forming expectations about the future which is why we choose this general formulation. For convenience, we will continue to refer to w0 as current wealth, keeping in mind that this figure may include expectations regarding future wealth levels. Our purpose in this section is to provide a precise definition of measures that capture the dependence of insecurity on past wealth movements in addition to today’s wealth level. Loosely speaking, the basic hypothesis is that insecurity increases (decreases), ceteris paribus, with decreases (increases) of the wealth level experienced in the past with higher weight given to the recent past. Thus, we propose to use the following three axioms as the defining properties of an individual measure of economic insecurity, provided that the observed variable is individual net wealth. Single-period monotonicity. V 0 is a decreasing function of w0 . The interpretation of single-period monotonicity is straightforward and intuitive: in the absence of any information regarding past wealth levels, insecurity is inversely related to current wealth. Difference monotonicity. For all T ∈ N, for all w ∈ R(T −1) and for all γ ∈ R, V T (w−(T −1) + γ, w) ≥ V T −1 (w) ⇔ γ ≥ 0. 4

Difference monotonicity requires a decrease in insecurity as a consequence of the ceteris paribus addition of another period −T which introduces a gain between periods −T and −(T − 1), thus allowing past gains to work against insecurity. Analogously, the measure of insecurity is assumed to increase if a period −T is added in a way such that wealth decreases, ceteris paribus, when moving from −T to −(T − 1). Finally, if the addition of period −T involves a wealth level identical to that of period −(T − 1), insecurity is unchanged. This is a monotonicity requirement that appears to be essential in capturing the notion of increased (decreased, unchanged, respectively) insecurity as a response to additional losses (additional gains, no changes, respectively) in past wealth levels. Note that the axiom does not imply that gains and losses have to be treated symmetrically; it is possible, for instance, that adding a gain of a certain magnitude, ceteris paribus, decreases insecurity by less than a loss of the same magnitude increases insecurity. We will return to this issue in more detail in section 4. The axiom is illustrated in figure 2. Starting from w0 = (3, 3, −1, 0, 2) ∈ R(4) , an additional period −5 is added to arrive at the stream w = (1, 3, 3, −1, 0, 2) ∈ R(5) . The move from period −5 to period −4 involves a gain in net wealth and, thus, difference monotonicity demands that V 5 (w) < V 4 (w0 ). wt

3

•

•

.......................................................... ... ... ... ... ... ... . . ... .. ... . ... ... ... ... . ... ... ... ... ... .. . ... . ... .. ... .. . . . . ... .. ... ... ... . ... ... ... ... .. ... . . ... ... ... ... ... ... . .. ... ... ... . . ... ... ... ... ... .. ... .. ... . . . ... ... ... ... ... ... ... .. ... . . ... ... ... ... ... .... ... ... . . . ... .... ... ... ... .... ... .... ... ... . . .. ... .... ... ... ... ....... ... .... . .... .. ..

2

1

0

•

•

−5 −1

−4

−3

−2

• −1

•

Figure 2: Difference monotonicity.

5

0

t

Proximity property. For all T ∈ N \ {1}, for all w ∈ R(T ) and for all τ ∈ {1, . . . , T − 1}, V T (w−T , . . . , w−(τ +1) , w−(τ +1) , w−(τ −1) , . . . , w0 ) ≥ V T (w−T , . . . , w−(τ +1) , w−(τ −1) , w−(τ −1) , . . . , w0 ) w−(τ +1) ≥ w−(τ −1) .

⇔

The proximity property ensures that a gain (loss) of a given magnitude reduces (increases) insecurity, ceteris paribus, to a higher extent the closer to the present this gain (loss) occurs. That is, changes in wealth from one period to the next have a more severe impact the closer they are to the present period. Figure 3 illustrates the axiom. Comparing the streams w = (1, 3, 3, −1, 0, 2) ∈ R(5) and w0 = (1, 3, −1, −1, 0, 2) ∈ R(5) , we see that w0 can be obtained from w by shifting the drop from 3 to −1 one period further into the past. According to the proximity property, the earlier loss affects insecurity to a lesser extent than the original one and, thus, insecurity in w0 is less than insecurity in w. wt

3

2

1

0

•

•

•

−5 −1

•

....................................................... ...... ... ... ... . ... ... .. .. ... ... . . ... . . ... ... ... . . . ... .. . ... ... .. . ... .. .. ... . . . ... . . ... ... ... . . . ... .. . ... ... ... .. .. . .. ... . .. . . . . ... . ... ... ... ... .. ... ... ... .. ... ... ... .. . .. . . . . .. ... .. ... ... .. ... ... ... ... .. ... ... ... . ... . . . . ... ... .. ... ... ... ... ... ... ... .. ... ... . ... ... ... ... .. ... ... .. ... ... .. . . ... . . ... ... ... ... ... .. ... ... ... . . . . . . ... ... .. .. ... .... ... .... ... ... .... .. ... ... . . . . ... ... .... ... ... .. ... .... ... ....... ... . . .. ....... ....... ....... ....... ......

−4

−3

−2

•

•

• −1

t

0

Figure 3: Proximity property. We suggest to use the above three axioms as the set of fundamental properties of a wealth-based measure of individual insecurity. This is parallel to the definition of an inequality measure as an S-convex function of individual incomes; see, for example, Dasgupta, Sen and Starrett [9] and Rothschild and Stiglitz [23].

6

3

Two-sequences Ginis

The Atkinson-Kolm-Sen approach to linking inequality to social evaluation was introduced by Kolm [17], independently suggested by Atkinson [1] and further investigated by Sen [24]. Given a strictly inequality averse social evaluation function, a corresponding inequality measure is obtained as one minus the ratio of equally-distributed-equivalent income and average income (the inequality index value is not defined if all incomes are equal to zero, although the social evaluation function can be defined on all possible distributions; in income inequality measurement, incomes are typically assumed to be non-negative and the zero vector is not included in their domain). An alternative approach is due to Kolm [17]. For any strictly inequality averse social evaluation function, Kolm [17] defines inequality as the difference between average income and equally-distributed-equivalent income. Blackorby and Donaldson [4] and Ebert [12] propose an ordinal approach to linking inequality and social evaluation. The Atkinson-Kolm-Sen inequality index is relative if and only if the equally-distributedequivalent income is a linearly homogeneous function of the income distribution. Moreover, the Kolm index is an absolute index if and only if the social evaluation function is unit translatable. The Gini [14] social evaluation function is well-established in the measurement of income inequality and in rank-ordered models of choice under risk; see, for instance, Rothschild and Stiglitz [21, 22, 23], Dasgupta, Sen and Starrett [9] and, more recently, Ben Porath and Gilboa [2] and Weymark [28]. An interesting feature of the Gini social evaluation function is that it is both linearly homogeneous and unit translatable and, thus, consistent with both a relative and an absolute measure of inequality. The Gini has a linear structure in rank-ordered subspaces where, for an n-person society, the coefficients of this linear function (from highest to lowest rank in a given income distribution) are the first n odd natural numbers. Mehran [19] and Weymark [27] independently introduced the generalized Ginis. These measures retain the linear structure of the Gini but allow for alternative degrees of inequality aversion by generalizing the coefficients to any rank-ordered sequence of parameters. As is the case for the Gini itself, the generalized Ginis are both linearly homogeneous and unit translatable. A subclass of the generalized Ginis is given by the single-series Ginis. They are generalized Ginis such that the sequence of coefficients is the same for all population sizes. See, for instance, Donaldson and Weymark [10, 11], Bossert [6] and Zank [29] for further observations regarding the generalized and the single-series Ginis.

7

Detailed discussions and formal definitions pertaining to the above-described approaches to inequality measurement can be found in Chakravarty [7] and Blackorby, Bossert and Donaldson [3]. We now characterize a specific class of measures that are inspired by the single-series Ginis. Although our characterization bears some resemblance to that of Bossert [6], there are substantial differences because we work with a different domain and axioms that are suited to the environment discussed here. A single sequence of coefficients no longer is sufficient to adequately express the notion of insecurity that we intend to capture here. Moreover, because we examine wealth levels assigned to time periods rather than to individuals, many properties and methods that are appropriate in the context of inequality or risk measurement are not suitable in our setting. A detailed discussion of some important features that distinguish insecurity measurement from inequality or risk measurement is provided in section 5. The class of two-sequences Gini measures involves two sequences of parameters—one sequence that applies to past period-to-period losses in wealth and one sequence that is used for period-to-period gains. The sequences need not be the same but, within each sequence, some natural restrictions apply. Let α = hα−t it∈N and β = hβ−t it∈N be two sequences of parameters such that   α−t > α−(t+1) > 0 and β−t > β−(t+1) > 0

for all t ∈ N.

(1)

The set of all sequences α such that α−t > α−(t+1) > 0 for all t ∈ N is denoted by C. C 2 is the Cartesian product of C with itself, that is, C 2 is the set of all pairs of sequences (α, β) satisfying (1). The two-sequences E measure of insecurity corresponding to a pair of D Gini T sequences (α, β) ∈ C 2 , V(α,β) = V(α,β) , is defined by letting, for all T ∈ N0 and for T ∈N0 all w = (w−T , . . . , w0 ) ∈ R(T ) , T V(α,β) (w) =

X


α−t w−t − w−(t−1) +

t∈{1,...,T }: w−t >w−(t−1)

X


β−t w−t − w−(t−1) − w0 .

t∈{1,...,T }: w−t
For an illustration of the two-sequences Gini measures, consider figure 4. We use the 1 parameter values α−1 = 1 and β−1 = 1/2 and depict the level sets of V(1,1/2) corresponding

to the levels 1 (the piecewise linear set farthest to the northwest), 0 (the middle piecewise linear set) and −1 (the piecewise linear set farthest to the southeast).

8

w−1

4

... ... ... ... ... ... .. .. . . ... ... ... ... ... ... .. .. . . . . ... ... ... ... ... ... . .. .. . ... . ... ... ... .. ... ... .. . .. .. . . . . ... ... ... ... ... ... ... ... ... .. ... .. . . . ... ... ... ... ... ... .. ... ... .. .. .. . .. . . . . . . ... ... ... ... .. .. . ... . . . . . .. .. ... ... .. ... ... ..... ... ... ... ... .. ..... .. .. . . . ...... ... ... ... ... ... .. .. .. ... .. .. ..... . . . . . . .. .. ... ... .... .. ... ... . .. ... ... .... .... .. . ... .. . . . ... ... ... ... ... ... ... ... ... .. ... ... .. .. .... .. . . . .. . ... .... ... ... .... ... ... ... ... .. .. ....... . . . ... ... .. ... .... ... ... ... ... ... ... ..... .. .. .. . . . ... ... ..... ... ... ... ... .. .. ... .... .. .. .... . . . . . . . .. ... ... ... . .. ... ... .... ... .. .. .. .. . .... . . . . . . . . .. ... ... . .. ... .... ... ... ... . ... ... .. .. .... .. . . . .. ... ... ... ... ... ... ... .. ... ... ... .... .. .. .. . . . ... ... ... ... ... ... ... ... ... .. .. .. . . . ... ... ... ... ... ... ... ... ... .. .. .. . . . ... ... ... ... ... ... ... ... ... .. .. .. . . . ... ... ... ... ... ... ... ... ... .. .. .. . . . ... ... ... ... ... ... .. .. .. .. .. .. . . . . . . .. .. ..

3

2

1

−2

−1

1

w0

2

−1

−2

−3

−4

1 Figure 4: The level sets of V(1,1/2) for the values 1, 0 and −1.

In addition to the defining properties of an individual measure of insecurity introduced in the previous section, the following axioms are used to characterize the class of twosequences Gini measures of individual insecurity. For r ∈ N, we use 1r to denote the vector consisting of r ones. Homogeneity. For all T ∈ N0 , for all w ∈ R(T ) and for all λ ∈ R++ , V T (λw) = λV T (w).

9

Homogeneity is a standard requirement that demands insecurity to be measured by means of a ratio scale. Translatability. For all T ∈ N0 , for all w ∈ R(T ) and for all δ ∈ R, V T (w + δ1T +1 ) = V T (w) − δ. Translatability differs from the usual translation scale property in that the value of δ is subtracted from the level of insecurity when δ is added to the wealth level in each period. This is a consequence of the inverse relationship between wealth and insecurity. The conjunction of homogeneity and translatability implies a strengthening of singleperiod monotonicity, as stated in the following lemma. Lemma 1. If a measure of individual insecurity V satisfies homogeneity and translatability, then V 0 (w0 ) = −w0

for all w0 ∈ R.

(2)

Proof. Setting T = 0 and w0 = 0, homogeneity implies V 0 (0) = V 0 (λ · 0) = λV 0 (0)

for all λ ∈ R++

and, substituting any λ 6= 1, it follows that V 0 (0) = 0.

(3)

Setting T = 0 and δ = −w0 in the definition of translatability and using (3), we obtain V 0 (0) = V 0 (w0 + (−w0 )) = V 0 (w0 ) + w0 = 0

for all w0 ∈ R.

(4)

Clearly, the last equality in (4) is equivalent to (2). Note that the full force of homogeneity and translatability is not needed for the above lemma; as is evident from the proof, it is sufficient to use the respective properties that are obtained by restricting the scopes of the axioms to the cases in which T = 0. Temporal aggregation property. For all T ∈ N\{1}, there exists a function ΦT : R2 → R such that, for all w ∈ R(T ) ,
V T (w) = ΦT w−T − w−(T −1) , V T −1 (w−(T −1) , . . . , w0 ) . 10

The temporal aggregation property is a separability condition that allows a measure of insecurity to be calculated by recursively moving back from the current period to the earliest relevant period where, in the step involving period −t, the part of insecurity that takes into consideration all periods from −t to the current period is obtained as an aggregate of the insecurity resulting from considering periods −(t − 1) to period zero only and the change experienced in the wealth level between periods −t and −(t − 1); see Blackorby, Primont and Russell [5] for a detailed discussion of various recursivity properties. If added to the basic requirements of the previous section (except for single-period monotonicity which is redundant; see lemma 1), these axioms characterize the class of two-parameter Gini measures of insecurity. We obtain Theorem 1. A measure of individual insecurity V satisfies difference monotonicity, the proximity property, homogeneity, translatability and the temporal aggregation property if and only if there exists (α, β) ∈ C 2 such that V = V(α,β) . Proof. ‘If.’ Let (α, β) ∈ C 2 . That V(α,β) satisfies homogeneity and translatability is immediate. Difference monotonicity follows from the positivity of the coefficients α−t and β−t ; see the definition of C. The proximity property is satisfied because of the inequalities that apply to the sequences of parameters; see, again, the definition of C. To see that the temporal aggregation property is satisfied, define, for all T ∈ N \ {1}, the function ΦT : R2 → R by letting, for all (x, y) ∈ R2 ,    α−T x + y T Φ (x, y) = y   β−T x + y

if x > 0 if x = 0 if x < 0.

‘Only if.’ Suppose V satisfies the required axioms. We prove the relevant implication by inductively constructing a pair of sequences (α, β) ∈ C 2 such that V T (w) =

X


α−t w−t − w−(t−1) +

t∈{1,...,T }: w−t >w−(t−1)

X


β−t w−t − w−(t−1) − w0

(5)

t∈{1,...,T }: w−t
for all T ∈ N0 and for all w ∈ R(T ) . If T = 0, (5) is satisfied for all w = (w0 ) ∈ R(0) (trivially, for any pair (α, β) ∈ C 2 and, in particular, for the pair of sequences to be constructed below) because of (2). Now let T = 1.

11

If w ∈ R(1) is such that w−1 = w0 , difference monotonicity and (2) together imply V 1 (w) = V 0 (w) = −w0 .

(6)

If w is such that w−1 > w0 , translatability with δ = −w0 implies V 1 (w−1 − w0 , 0) = V 1 (w−1 − w0 , w0 − w0 ) = V 1 (w−1 , w0 ) + w0 = V 1 (w) + w0 and, therefore, V 1 (w) = V 1 (w−1 − w0 , 0) − w0 .

(7)

Applying homogeneity with λ = w−1 − w0 > 0, it follows that V 1 (w−1 − w0 , 0) = V 1 ((w−1 − w0 ) · 1, (w−1 − w0 ) · 0) = (w−1 − w0 )V 1 (1, 0) and, together with (7), V 1 (w) = α−1 (w−1 − w0 ) − w0

(8)

where α−1 = V 1 (1, 0). By difference monotonicity, α−1 > 0. If w is such that w−1 < w0 , translatability with δ = −w0 implies V 1 (w−1 − w0 , 0) = V 1 (w−1 − w0 , w0 − w0 ) = V 1 (w−1 , w0 ) + w0 = V 1 (w) + w0 and, therefore, V 1 (w) = V 1 (w−1 − w0 , 0) − w0 .

(9)

Applying homogeneity with λ = −(w−1 − w0 ) > 0, it follows that V 1 ((w−1 − w0 ), 0) = V 1 (−(w−1 − w0 ) · (−1), −(w−1 − w0 ) · 0) = −(w−1 − w0 )V 1 (−1, 0) and, together with (9), V 1 (w) = β−1 (w−1 − w0 ) − w0

(10)

where β−1 = −V 1 (−1, 0). By difference monotonicity, β−1 > 0. Combining (6), (8) and (10), we obtain X
V 1 (w) = α−t w−t − w−(t−1) + t∈{1}: w−t >w−(t−1)

X


β−t w−t − w−(t−1) − w0

t∈{1}: w−t
for all w ∈ R(1) . Now suppose that T ∈ N \ {1} and X
V T −1 (w) = α−t w−t − w−(t−1) + t∈{1,...,T −1}: w−t >w−(t−1)

X t∈{1,...,T −1}: w−t
12


β−t w−t − w−(t−1) − w0 (11)

for all w ∈ R(T −1) where (α−(T −1) , . . . , α−1 ) and (β−(T −1) , . . . , β−1 ) are such that α−1 > . . . > 0 and β−1 > . . . > 0. We have to show that there exists (α−T , β−T ) such that α−1 > . . . > α−T > 0 and β−1 > . . . > β−T > 0

(12)

and X

V T (w) =

X


α−t w−t − w−(t−1) +

t∈{1,...,T }: w−t >w−(t−1)


β−t w−t − w−(t−1) − w0

(13)

t∈{1,...,T }: w−t
for all w ∈ R(T ) . Together with (11), the temporal aggregation property implies the existence of a function ΦT : R2 → R such that
V T (w) = ΦT w−T − w−(T −1) , V T −1 (w−(T −1) , . . . , w0 )  X
α−t w−t − w−(t−1) = ΦT w−T − w−(T −1) , t∈{1,...,T −1}: w−t >w−(t−1)

X

+


β−t w−t − w−(t−1) − w0

(14)

t∈{1,...,T −1}: w−t
for all w ∈ R(T ) . First, consider w ∈ R(T ) such that w−T = w−(T −1) . Difference monotonicity and (11) together imply V T (w) = V T −1 (w−(T −1) , . . . , w0 ) X
= α−t w−t − w−(t−1) + t∈{1,...,T −1}: w−t >w−(t−1)

X


β−t w−t − w−(t−1) − w0

t∈{1,...,T −1}: w−t
and it follows that ΦT (0, y) = y

for all y ∈ R2 .

(15)

Now consider the case in which w is such that w−T > w−(T −1) . Homogeneity implies that ΦT satisfies ΦT (λx, λy) = λΦT (x, y)

for all λ, x ∈ R++ and for all y ∈ R

(16)

and translatability implies ΦT (x, y − δ) = ΦT (x, y) − δ

for all x ∈ R++ and for all δ, y ∈ R. 13

(17)

Letting δ = y, (17) implies ΦT (x, 0) = ΦT (x, y) − y and, thus, ΦT (x, y) = ΦT (x, 0) + y

for all x ∈ R++ and for all y ∈ R.

(18)

Letting λ = x > 0, (16) implies ΦT (x, 0) = ΦT (x · 1, x · 0) = xΦT (1, 0)

for all x ∈ R++

and, together with (18), we obtain ΦT (x, y) = α−T x + y

for all x ∈ R++ and for all y ∈ R

(19)

with α−T = ΦT (1, 0). By difference monotonicity, α−T > 0 and by the proximity property, α−T < α−(T −1) and, thus, α−1 > . . . > α−T > 0.

(20)

Finally, suppose w is such that w−T < w−(T −1) . Homogeneity implies that ΦT satisfies ΦT (λx, λy) = λΦT (x, y)

for all λ ∈ R++ , for all x ∈ R−− and for all y ∈ R

(21)

and translatability implies ΦT (x, y − δ) = ΦT (x, y) − δ

for all x ∈ R−− and for all δ, y ∈ R.

(22)

Letting δ = y, (22) implies ΦT (x, 0) = ΦT (x, y) − y and, thus, ΦT (x, y) = ΦT (x, 0) + y

for all x ∈ R−− and for all y ∈ R.

(23)

Letting λ = −x > 0, (21) implies ΦT (x, 0) = ΦT (−x · (−1), −x · 0) = −xΦT (−1, 0)

for all x ∈ R−−

and, together with (23), we obtain ΦT (x, y) = β−T x + y

for all x ∈ R−− and for all y ∈ R

(24)

with β−T = −ΦT (−1, 0). By difference monotonicity, β−T > 0 and by the proximity property, β−T < β−(T −1) and, thus, β−1 > . . . > β−T > 0. Combining (15), (19) and (24), it follows that    α−T x + y T Φ (x, y) = y   β−T x + y 14

if x > 0 if x = 0 if x < 0

(25)

for all (x, y) ∈ R2 . Substituting back into (14), we obtain (13) for all w ∈ R(T ) . Because α−1 > 0 and β−1 > 0 and, moreover, (20) and (25) are satisfied for all T ∈ N \ {1}, the pair of sequences (α, β) thus constructed satisfies (12) and therefore is in C 2 as required.

The definition of the two-sequences Ginis is silent about the relationship between the current wealth level and the coefficients α−1 and β−1 applied to the most recent gains and losses. This lack of specification is intentional: it is not obvious to us how current levels are to be traded off against past differences. Among other things, this may depend on the interpretation of w0 and, particularly, on whether this variable represents current wealth only or an aggregate of current wealth and future expectations on the part of an agent.

4

Two-sequences rank-ordered Ginis

The theorem of the previous section does not impose any restrictions on the relationship between the sequences α and β. A plausible assumption appears to be the requirement that ceteris paribus losses of a certain magnitude in a given period have at least as strong an impact on insecurity as ceteris paribus gains of the same magnitude in the same period. This assumption is captured in the following axiom. Weak loss priority. For all T ∈ N, for all w ∈ R(T −1) and for all γ ∈ R++ , V T (w−(T −1) + γ, w) − V T (w−(T −1) , w) ≥ V T (w−(T −1) , w) − V T (w−(T −1) − γ, w).

Weak loss priority can be interpreted as an insecurity analogue of weak risk aversion in the context of individual choice under uncertainty. Figure 5 provides an example of the application of this axiom. The three wealth streams indicated differ in the earliest period (period −5) only. The uppermost stream starts at a net wealth level of 5, the second at 3 and the third at 1. Thus, the absolute value of the difference between the first and the second stream in this period is the same as the absolute difference between the second and the third stream (this difference is given by 2). The distinguishing feature between these differences is that the first represents a loss while the second is a gain with respect to the middle stream. Weak loss priority requires that the loss has a larger impact on insecurity than the (equal-sized) gain so that V 5 (5, 3, 3, −1, 0, 2) − V 5 (3, 3, 3, −1, 0, 2) ≥ V 5 (3, 3, 3, −1, 0, 2) − V 5 (1, 3, 3, −1, 0, 2). 15

wt

5

•.. . . . . . . . .

4

. . . . . .

3

2

1

0

. . . ....... ....... ....... ....... ............................................................ ... ... ... .. . ... ... ... .. ... . .. ... . .. ... . ... .. . . ... . . ... ... . ... .. . ... .. . ... ... .. ... . ... .. .. ... . .. ... .. . . . . ... ... ... ... ... ... ... .. ... ... .. .. . . . . . ... ... ... ... ... ... ... .. ... ... .. ... . . . ... ... ... ... ... .. ... .. . . ... .. ... .. ... ... ... ... ... .. . ... . ... ... .... ... .... ... ... . . ... . ... ... ... ... .... ... ... ... .... . . . ... .... ... ... ... ...... ... ..... .....

•

•

•

•

−5 −1

•

−4

−3

−2

• −1

t

0

•

Figure 5: Weak loss priority. Let D be the set of all pairs of sequences (α, β) ∈ C 2 such that α−t ≥ β−t

for all t ∈ N.

(26)

A two-sequences Gini insecurity measure V(α,β) is rank-ordered if (α, β) ∈ D. Adding weak loss priority to the axioms of theorem 1 leads to a characterization of the rank-ordered two-sequences Gini measures. That the rank-ordered two-sequences Gini measures satisfy weak loss priority is an immediate consequence of (26). The proof of the only-if part of the following theorem follows from theorem 1 and the observation that weak loss priority implies (26), given that V is a two-sequences Gini. Theorem 2. A measure of individual insecurity V satisfies difference monotonicity, the proximity property, homogeneity, translatability, the temporal aggregation property and weak loss priority if and only if there exists (α, β) ∈ D such that V = V(α,β) . Theorem 2 identifies the class of insecurity measures that we advocate in this paper. As an example, consider the measure obtained by choosing the sequences α and β so that 1 α−t and β−t = α−t = 2t − 1 2 16

for all t ∈ N. Clearly, (α, β) ∈ D. The coefficients according to the sequence α are the inverses of the coefficients corresponding to the Gini social evaluation function; see, for instance, Donaldson and Weymark [10] and Weymark [27].

5

Insecurity, inequality and risk

The two-sequences (rank-ordered) Gini measures of insecurity share a linear structure with the generalized Ginis used in inequality measurement and some rank-ordered decision criteria employed in theories of choice under uncertainty. However, despite these similarities, there are some significant differences due to the nature of the variables considered and the phenomenon to be measured. The coefficients of the single-series Gini social evaluation functions are applied to welfare-ranked (from highest to lowest) income distributions. There is a natural dual class, namely, the illfare-ranked (from lowest to highest) single-series Ginis, where a single sequence of coefficients is applied to illfare-ranked income vectors. There is no analogue of this dual class in the insecurity setting. This is the case because our variables are wealth levels over time rather than income or wealth levels of different individuals. As a consequence, there is no natural first period to be taken into consideration. The natural reference period is the current period and wealth streams with histories of varying lengths have to be compared. For that reason, the temporal aggregation property employed in our theorems can be formulated in one direction only—as a recursion from the current period backwards. Donaldson and Weymark [10] use Dalton’s [8] principle of population to narrow down the class of single-series Ginis to a single-parameter class. The principle of population applies to replications of a society and demands that, for a given income distribution, if all incomes are replicated, inequality and equally-distributed-equivalent income remain unchanged. Replications are not as natural in the insecurity context. The reason is that wealth levels in different periods are replicated and the different weights of the twosequences Ginis depend on the periods rather than the rank order of the wealth levels. If a the wealth stream w is replaced with an r-fold replication w1r with r ∈ N \ {1}, gains and losses are introduced at different periods the net effect of which may be positive, negative or zero. For instance, consider w = (w−1 , w0 ) and w12 = (w−1 , w0 , w−1 , w0 ). Current wealth continues to be w0 but we have added two gains or losses: the difference w0 −w−1 is replicated further in the past and, more importantly, the difference w−1 −w0 is added between periods −1 and −2. Clearly, the overall effect on insecurity of such a move 17

cannot be determined in general. Analogously, if w is replicated period-by-period to obtain the stream (w−T 1r , . . . , w0 1r ), the resulting change in insecurity cannot be determined without further knowledge. This is due to the observation that the treatment of the current wealth level w0 differs from the treatment of all earlier wealth levels, which only enter in the form of gains or losses into the value of a two-sequences Gini. Moreover, the direction of change depends on the initial value of the insecurity measure at w. The above observations suggest that the approach to insecurity measurement proposed in this paper is not a mere reformulation of known results in the theory of risk and in inequality measurement; it deals with a new class of measures that, in spite of some formal similarities, have quite different properties and interpretations. Our measures are sufficiently flexible to admit the use of any model describing the formation of individual expectations. We intentionally do not pick a specific theory of decision-making under uncertainty in order to allow for a wide variety of possible applications. The exploration of such applications constitutes a natural next step in this area of research.

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