The Inequality Multiplier by Columbia Global Policy Initiative

The Inequality Multiplier Gallegati M.1 , Landini S.∗2 , and Stiglitz J.E.3 1

DiSES, Department of Economics and Social Sciences, Faculty of Economics ”Giorgio Fuà”, Università Politecnica delle Marche, Piazzale Martelli, 8, 60121, Ancona, Italy. 2 I.R.E.S. Piemonte, Istituto di Ricerche Economico Sociali del Piemonte, Via Nizza, 18, 10125, Turin, Italy. 3 Columbia Business School, Graduate School of Arts and Sciences and the School of International and Public Affairs, 3022 Broadway, New York, NY 10027, USA.

Abstract That of the multiplier is a largely debated issue. Several studies propose estimates for it. This paper answers the question of how inequality affects the value of the multiplier. The proposed formulation is analytically derived from the Lorenz curve of income by means of Zanardi asymmetry index. Since the relationship between inequality and multiplier is found to be negative, it can be argued that greater inequality has depressive effects on GDP.

Keywords: Income multiplier, Income distribution, Lorenz curve asymmetry, Inequality.

JEL classification: B41, C65, D31, D63.

∗ Corresponding

author: landini@ires.piemonte.it

Electronic copy available at: http://ssrn.com/abstract=2766301

Contents 1 Introduction

2 Basics on the Lorenz curve 2.1 Some properties of the Lorenz curve . . . . . . . . . . . . . . . . 2.2 The importance of being asymmetric . . . . . . . . . . . . . . . .

6 7 7

3 Asymmetry of the Lorenz curve: an inequality measure 3.1 The Gini transform of the Lorenz curve . . . . . . . . . . 3.2 The Zanardi asymmetry index . . . . . . . . . . . . . . . 3.3 Characteristic points . . . . . . . . . . . . . . . . . . . . . 3.4 The LGZ methodology to measure inequality . . . . . . .

. . . .

11 11 12 15 17

4 Asymmetry and heterogeneity, as if they were important

5 The inequality multiplier

6 Concluding remarks

Electronic copy available at: http://ssrn.com/abstract=2766301

Introduction

During the two decades prior to the beginning of the global economic crisis, the gap between rich and poor grew in many of the G7 countries ([17],[20],[23]). From the late eighties to mid-nineties, the increase in income inequality was particularly accentuated in the United Kingdom, the United States and Canada. Those years marked a growing gap between rich and poor, also - for the first time - in countries with traditionally low income inequality, such as Germany and Italy. In emerging countries, economic growth plays a decisive role in reducing poverty, but at the same time, income distribution becomes more concentrated. Stliglitzâ&#x20AC;&#x2122;s hypothesis is that inequality is not only the result of abstract market forces, but the result of government policies that shape and drive the forces of technology and markets, as well as, in a wider sense, of society as a whole. In a vicious circle, greater inequality has led to less equality of opportunity, and therefore to greater inequality. Growing inequality jeopardizes support for collective action, which is what guarantees that everyone can fully develop his/her full potential, as a result, for example, of investment in public education. Also, inequality fosters instability, which in turn gives rise to more inequality. Relocating money to high-income individuals, who in proportion consume a smaller slice of their earnings than the poor, reduces consumption and the multiplier (richer people save 15-25 percent of their income, while the poor spend nearly all of it). The result is that until something happens, like for example an increase in investments or exports, the total demand will be lower than what the economy would be able to meet, i.e. there will be unemployment. In the nineties that something was the technology bubble; in the first decade of the twenty-first century there was the housing bubble. Now the only solution is to resort to a redistribution policy and/or public spending. Unemployment today can be attributed to a shortage arising from aggregate demand (total demand for goods and services in economy by consumers, firms, government and exporters); in a certain sense, the entire current deficit in aggregate demand is due to phenomena of inequality, which is approaching unsustainable limits for societies. Maybe the great depression was not caused by the unequal distribution, but the effects on income via the multiplier have been quite relevant. But what is the value of the multiplier? There are several estimates about it: the Journal of Economic Literature published on September 2001 a symposium on the size of the multiplier, [2]. Most models of the US economy put the multiplier at between 0.8 and 1.5. A value just greater (0.9 â&#x2C6;&#x2019; 1.7) than that has been identified by Blanchard and Leigh [4] for 26 economies. In the present paper, the proposed analytical framework of the multiplier takes into account inequality. In order to evaluate the effects of inequality on the income multiplier and GDP it takes a suitable estimator for propensity to consume, that is an estimator which takes care of heterogeneity, concentration and asymmetry of the income distribution. Therefore, this note aims at integrating information from the income distribution into macroeconomic modeling [3], without explicitly considering either the modeling of the Dalton-Pigou ([5],[19]) and the mean-preserving spread [22] 3

principles effects. The identified multiplier goes beyond the averaging assumptions of the tradition and explains some of the effects of inequality in income distribution on the multiplier by developing an inequality adjusted formulation of it by considering heterogeneity in income distribution, propensity to consume, income concentration and, above all, the asymmetry of the distribution. The starting point is fixed in the following textbook-like Proposition 1 In a closed economy without Government, a decrease in the marginal propensity to consume decreases ( ceteris paribus with respect to inequality and concentration) the value of the Keynesian multiplier. This means that if the distribution of income is positively skewed, as it often happens, an increase of inequality in income distribution, usually detected by an increase of the Gini index [10], makes the propensity to consume decreasing so lowering the income multiplier. But, as known, the Gini index measures only the concentration of income and, in general, it does not allow for a direct comparison of two Lorenz curves (see [8], [14]), mainly if they intersect with each other [1]: one can be consistent with a Lorenz curve asymmetric toward the poorest classes, while the other toward the richest ones, just like represented in Figure 2. To overcome this methodological problem, a measure of asymmetry of the Lorenz curve is involved while taking care of income heterogeneity and concentration. Although that of inequality is a largely debated topic since ever1 , it is opportune since now recalling what inequality is meant to mean to the purposes of the present paper. Broadly speaking, in general a given non-negative, limited, transferable quantity W , semantically related to some positive meaning of enjoyment, is said to be characterized by inequality if it unfavorably distributes over a wide spectrum of receivers while favorably distributing over a restricted spectrum of other ones. Three remarks are now worth stressing. First of all it should be observed that, apart from the rectangular (i.e. Uniform) distribution case, every non-negative, limited and transferable quantity non-uniformly distributes among its receivers, hence inequality is the regular case. Secondly, the notion of inequality is not a homogeneous constant: i.e. different samples of W in space and/or time can be characterized by different degrees of inequality, that depends on how wide the spectrum of the (un-)privileged receivers is. Finally, that of inequality is not an absolute: depending on the specific point of view, what is favorable for those receivers who get an advantage is unfavorable for those who get a disadvantage from the same distribution. In the present paper, the unprivileged point of view is assumed. Accordingly, what in the following is labeled as the poor side is always disadvantaged with respect to the rich side complement: of course, the poor side may be more or less disadvantaged in different situations but, by definition, the rich side is always advantaged. 1 And there may be wide differences of opinion as to the significance of a very unequal distribution, [14][p.1]. But, the objection to great inequality of incomes is the resulting loss of potential economic welfare, [5][p.349].

Being W income it immediately comes clear the relevance of this generic interpretation of inequality. Income is not only non-negative, transferable and limited within [wmin ; wmax ], it is also semantically correlated to some positive meaning: in a sense, the more the income one earns the better it is. Therefore, the more the income one earns the higher the advantage she owes to the distribution of income, that is because it would give her more opportunity of enjoyment in consumption or investment. As known, a restricted share (1 − pd ) of privileged consumers is above a given threshold wd of income, and it accumulates a share (1−qd ) of total income which is larger than the small share qd accumulated by the complementary large share pd of unprivileged consumers. That is, there are but a few consumers sharing a lot of total income and a lot of consumers sharing a few of it. This implies that the ratio (1 − qd )/(1 − pd ) is higher than qd /pd hence, on average, the privileged per-capita income is greater than the unprivileged one, as it could not be otherwise. This view approaches to the inequality phenomenon to refer to in the following, and it stems to three main aspects synthesized in the following Definition 1 A positive and transferable quantity with diversification of advantages in opportunity of enjoyment among its receivers is unequal because its distribution is heterogeneous, concentrated and asymmetric. Being inequality the regular case, Definition 1 should be considered as the null hypothesis, whose notions easily apply to the case of income: • heterogeneity: consumers can be split at least into rich and poor sides on the income distribution; i.e. although within each side there can be a certain degree of diversification, there exists at least one formal discriminant level of income separating the two sides; • concentration: the rich side is populated by few consumers sharing the hugest part of total income while the poor side is populated by a lot of consumers sharing a few of it hence, on average, the rich per-capita income is greater than the poor one; • asymmetry: the imbalance in income distribution is such that the diversification among rich consumers is higher than that among poor ones, which means the poor side is more within homogeneous and disadvantaged than the rich one. Each of these notions has its own counterpart - i.e. homogeneity, diffusion and symmetry - related to the ideal notion of equal distribution. Usually, inequality is limited to the concentration of the Lorenz curve because it means that, below some point in the distribution, some (poor) classes are precluded to the same level of enjoyment reserved to other (rich) ones: this is the opposite case of diffusion, which leads to equality in the opportunity of enjoyment, nicely correlated to some positive meaning, of course not for all and not always2 . Equality and 2 This

may be related to the Robin Hood paradox [13] and to the Matthew effect [16].

inequality are opposite states along a continuum of dis-advantaging enjoyment degrees for un-privileged receivers on the distribution of income: given a certain configuration, to account for any change in inequality it is proposed considering heterogeneity, concentration and asymmetry at once. Although recognizing that income inequality is the null hypothesis, the degree of inequality can be differentiated through time and space. Moreover, as the regular case, inequality cannot be completely removed to attain an uniform distribution of income among consumers, it nevertheless means nothing can be done to limit its increase beyond critical limits. Along with these lines, the macroeconomic perspective of this note warns that: (a) increasing inequality depresses consumption, (b) depressing consumption depresses the GDP. Hence, according to Definition 1, Proposition 1 generalizes to the following Thesis 1 If income distribution becomes more unequal the multiplier of income decreases, with all its consequences. This paper aims at sustaining Thesis 1 in three main steps, which structure the sectioning of the paper into parts as follows. The first part (sections 2 and 3) translates the notion of inequality into a measure based on an asymmetry index of the Lorenz curve of the distribution of income. The second part (sections 4 and 5) develops a model for the inequality adjusted marginal propensity to consumption curve associated to an income distribution, and derives the inequality adjusted multiplier of income. The last part (Section 6) concludes.

Basics on the Lorenz curve

This section briefly reports on some properties of the Lorenz curve. Assume income is a random variable W with support W = [wmin ; wmax ] on the real line, being 0 < wmin wmax < +∞. Let the distribution of income be known as F(w) = P{W ≤ w}, hence pw = F(w) is the cumulative share of consumers whose income is not above the level w Z w pw = F(w) = f (u)du : F(wmin ) = 0 , F(wmax ) = 1 , pw ∈ [0; 1] (1) wmin

such that w = F −1 (pw ) = inf{u : F(u) ≥ w} ∈ [wmin ; wmax ]

(2)

is the fractile associated to pw . The first incomplete moment of W at w is Z w F1 (w) = uf (u)du : F1 (wmax ) = µ (3) wmin

hence µ is the average. The analytic Lorenz curve [8] on the (p, q)-plane is qw = L(pw ) =

F1 (F −1 (pw )) , qw ∈ [0; 1] µ 6

(4)

as represented in Figure 1, and the Gini index is Z 1 G =1−2 L(p)dp

(5)

Therefore, the Lorenz curve describes how the cumulative share qw of total income changes on the vertical axes of the (p, q)-plane as the cumulative share pw of consumers changes on the horizontal axes. By evaluating the area above the L-curve and below the equi-distribution line q = p, as limited by the g in Figure 1, the Gini index provides a concentrasegment OE and the arc OE tion measure, that is it evaluates how much the distribution of W is far from the equi-distribution, therefore it accounts for an inequality indication, unfortunately without reference to who is concentrating the hugest share of W .3

2.1

Some properties of the Lorenz curve

Due to its definition in (4), L ∈ C 2 : [0; 1] → [0; 1] such that L(0) = 0 and L(1) = 1. Moreover, being differentiable it is continuous and, by definition, it is also monotonically increasing and convex: i.e. L0 (p) > 0, L00 (p) > 0. Accordingly, qw ≤ L(pw ) ∀pw = F(w): i.e. the share of total income accumulated by the first pw % of consumers cannot exceed the share of consumers below the income level w. For any L-curve there exists one and only one point D(pd , qd ) of intersection with the negative bisector q = 1 − p, also said the axes of symmetry: i.e. L(pd ) = qd = 1 − pd . For any L-curve there exists one and only one point C(pc , qc ) at which the tangent line to the curve is parallel to the equi-distribution bisector q = p: i.e. L0 (pc ) = 1 s.t. L(pc ) = qc . As a consequence of these properties, to any pw = F(w) on the (w, p)-plane one and only one qw = L(pw ) is associated on the (p, q)-plane. Therefore, the L-curve on the (p, q)-plane, implied by the distribution F on the (w, p)-plane, provides a description of W which is equivalent to that of the distribution F, but the contrary is not always true. Therefore, the association F ⇒ L is unique but the contrary is not guaranteed.

2.2

The importance of being asymmetric

One of the main consequences of previous properties is that two Lorenz curves may have the same G index although performing different or even opposite shapes, see Figure 2. Hence, beyond the concentration degree, knowing the Gini index tells almost nothing about the shape of the L-curve and of the distribution F. On the contrary, if one knows something about the shape of the L-curve then knowledge of the G index becomes more informative. 3 It is worth stressing that, given W with a known distribution F (w) an analytic Lorenz curve L(pw ) is uniquely implied, whilst an empirical estimate of the Lorenz curve can be consistent with different analytic distributions. On the other hand, given W a Gini index G is uniquely associated, whilst an empiric estimate of the concentration can be consistent with different income distributions, even when characterized with different imbalances.

As the regular case prescribes, income distribution is unequal in the sense that a few consumers share a lot of income while a lot of consumers share a few of it. According to Definition 1 this means consumers are heterogeneous, income concentrates more in a restricted set of consumers, and it also means there is some imbalance in distribution called asymmetry: these notions concur to the definition of inequality. Therefore, if this regularity persists through time and/or across space, then an increase of G may lead to a more unequal distribution, although this is not always guaranteed to be true. If G measures how much L is far from equi-distribution, there should exist some statistics about the shape of the L-curve to understand where does the imbalance come from: this is the general asymmetry index [6] S = F(w̄) + L(p̄) > 0

(6)

where w̄ is the average income and p̄ is the cumulative share of consumers not exceeding the average income. The statistics S relates the distribution F to the Lorenz curve L: this is possible because, as said, once F is given on the (w, p)-plane the implied L-curve is equivalent on the (p, q)-plane. A perfectly symmetric L-curve is such that the arc from the origin at O(0, 0) g has the same length and curvature of the arc to point D(pd , qd ), say OD, g connecting this intersection point with the upper extreme E(1, 1), say DE. Therefore, as far as F(w) = pw and L(pw ) = qw , expression (6) reads as S = p̄ + q̄ = 1 because, in general, q = 1 − p is the negative bisector the L-curve intersects. Hence, in general, the negative bisector reads as the axes of symmetry F(w) + L(pw ) = 1. The previously defined point C(pc , qc ) by construction lies on the L-curve at the maximum distance form the equi-distribution curve q = p. Therefore, in case of perfect symmetry, it comes clear that D ≡ C. This configuration is consistent with a perfectly symmetric distribution of income, but it does not mean concentration is null. Indeed, the point D may slide from the center point M0 (0.5, 0.5), at which G = 0, toward the bottom right corner M1 (1, 0), at which G = 1, and back. Therefore, if the L-curve is perfectly symmetric what matters is concentration. Unfortunately, inequality is the regular case and its definition finds evidence in the not perfectly symmetric cases. Hence, asymmetry should be considered at first and then concentration, while heterogeneity may be consistent either with the perfectly symmetric case and the asymmetric one. According to its definition, D(pd , qd ) ∈ M0 M1 is said a discriminant point4 , because it conventionally distinguishes the poor side of the L-curve, for which p ∈ [0; 0.5] and q ∈ [0; 0.5], from the rich one, for which p ∈ [0.5; 1] and q ∈ [0.5; 1]: it always exists and it is unique, hence it is the key point to introduce the minimal heterogeneity, regardless of concentration and asymmetry, and it is such that pd ∈ [0.5; 1] and qd ∈ [0; 0.5]. Nevertheless, the discriminant point 4 There may be several criteria to distinguish different parts on the L-curve, e.g. quartile points, median, mode and average points are valuable candidates. The here adopted one is easy to understand and, conventionally, introduces a minimal level of heterogeneity required by Definition 1.

is also relevant for asymmetry and concentration as well. Indeed, the ratios qd 1 − qd 0 ≤ Rdp = ≤ Rdr = ≤1 (7) pd 1 − pd depend upon D and measure, on a unit scale, the average per-capita income in the poor and rich sides respectively. Hence, the odds-ratio Od =

Rdr ≥1 Rdp

(8)

evaluates, on average, the proportion of a rich per-capita income compared to a poor one: i.e. how many poor per-capita incomes it takes to balance with a rich one. If Od = 1 it means Rdr = Rdp because qd = 1 − qd and pd = 1 − pd , which can happen only if D ≡ M0 , i.e. in a case of null asymmetry S = 0 and G = 0 concentration. If D → M1 then concentration is G → 1, hence Od → +∞ while asymmetry can be S > 1 if C is above the axis of symmetry q = 1 − p, or S < 1 if C is below. All in all, the discriminant point alone plays as the G index alone: except the degenerate case of perfect symmetry without concentration, which annihilates heterogeneity as in the case of perfect equality or uniform distribution, they both tell a few about the shape of the L-curve, moreover two different curves may share the same point D with the same G but different asymmetry profiles; see Figure 2. On the other hand, if properly considered and playing together, both the discriminant point and the Gini index are highly informative. According to its definition, C(pc , qc ) is said a critical point5 because it lies on the L-curve at the maximum distance from the equi-distribution line. Moreover, due to the above mentioned properties, it is worth noticing that such a critical point always exists and it is unique for a given Lorenz curve. Accordingly, if pc > pd then C lies above D then the L-curve performs a rightward imbalance toward the higher income classes: this means the curve is positively asymmetric and that the rich part is more concentrated than the poor one. Differently said, the rich part is more within heterogeneous than the poor one: in such a case income unfavorably distributes over the poor side in the sense that the poor consumers are more gradually poor while the rich ones are more increasingly rich while, by definition, they are sharing the hugest part of total income among a few. This is represented in Figure 1 and the solid (blue) line in Figure 2. In the opposite case, if pc < pd then C lies below D then the L-curve performs a leftward imbalance toward the lower income classes: this means the curve is negatively asymmetric and that the poor part is much more concentrated than the rich one. Therefore, the poor part is more within heterogeneous than the 5 As for the discriminant point, there may be several critical point criteria: the here adopted one makes it geometrically and analytically easy to find it, since it maximizes the distance of the L-curve from the equi-distribution line it pertains the definition of the Pietra-Ricci index (see [18], [21] and [7]). Another diffused criterion is based on the maximum curvature of the L-curve and leads to the Taguchi index [24].

L-curve 1

q w=L(p w): cumulative share of income

0.9

Critical point C(pc ,qc )

0.8 0.7 0.6

0.5

Discriminant point D(pd ,qd )

0.4

0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

p w=F(w): cumulative share of consumers

Figure 1: L-curve and characteristic points: data from [14], Prussia 1901. rich one: i.e. the income distribution unfavorably distributes over the rich side in the sense that the rich consumers are more gradually rich while the poor ones are more decreasingly poor while sharing the smallest part of total income among a few of consumers. This is represented by the (red) dashed line in Figure 2. Therefore, by involving both the discriminant D and the critical C points, all the concepts required by the notion of inequality adopted in Definition 1 and Thesis 1 are at hands. Figure 2 helps at reading the just described cases and it shows that two different Lorenz curves can pass through the same discriminant point with the same Gini index, although with different asymmetry profiles if they have different critical points. It is easy to imagine that two curves passing through the same discriminant and critical points are necessarily the same curve: this trait of uniqueness is due to previous properties. It also shows that the Gini index can measure inequality just in terms of concentration but it says nothing about the asymmetry, hence it is not suitable for a direct comparison, mainly if the curves intersect. Moreover, the figure shows that taking care of the asymmetry matters. Indeed, it may happen that at two different dates the Lorenz curve changes from L1 to L2 , maybe due to some critical event which twists the income distribution, but the concentration index does not grasp this phenomenon as the asymmetry index does. By measuring the concentration and the asymmetry of each sampling Lorenz curve it is possible to evaluate the inequality degree in different samples, they can be drawn from the same population at different dates, or from different populations at the same time or, finally, from different populations at different times as well.

L-curves with equal G and opposite imbalance 1

q w=L(p w): cumulative share of income

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

p w=F(w): cumulative share of consumers

Figure 2: Two Lorenz curves with the same Gini index but opposite imbalance profiles. The solid (blue) one through the circle C point has a positive asymmetry, the dashed (red) one through the diamond C point has a negative asymmetry. They share the same square D point.

Asymmetry of the Lorenz curve: an inequality measure

As shown, the direction of asymmetry of the Lorenz curve can be detected by the discriminant and critical points: it is positive if pc > pd and negative in the opposite case, while it is perfectly symmetric if pc = pd . Among other indexes available in literature, the degree of the asymmetry of the Lorenz curve can be measured by the Zanardi index (see [27, 28]).

3.1

The Gini transform of the Lorenz curve

To understand the meaning of the Zanardi asymmetry of the Lorenz curve it is useful introducing the Gini Γ-transformation (see [11] and [25, 26]). Let l = (p, q)0 be a point along the Lorenz curve on the plane P ≡ (p, q) (see I in Figure 3) as a vector space spanned by the canonical base I2 of R2 . Define 1 0 F= (9) 0 −1 operating a reflection of P w.r.t. the p-axes (see II in Figure 3) and find p r = Fl = −q

(10)

Vector r on P 0 is equivalent to l on P. Define then the matrix √ √ cos φ − sin φ 1/√2 −1/√2 Rφ = ⇒ R ≡ Rπ/4 = sin φ cos φ 1/ 2 1/ 2

(11)

which operates a counterclockwise rotation of P 0 by an angle φ = π/4 (see III in Figure 3) to get 1 p+q z = Rr = RFl = √ (12) p−q 2 on the reflected-rotated plane P 00 . Now consider an orientation preserving transformation on P 00 by multiplying a vector by a scalar γ√> 0: the vector is compressed if γ ∈ (0, 1) or stretched if γ > 1. Set γ = 2 as the length of the bisector q = p of the plane P (see I in Figure 3) and stretch z as follows p+q x g = γz = (γRF)l = Γl = = (13) p−q y which defines the Gini transformation x=p+q x 1 Γ: ⇔ = y =p−q y 1 whose inverse transform comes with Γ−1 p p = 21 (x + y) −1 ⇔ = Γ : q q = 12 (x − y)

1 2 1 2

1 −1

1 2 − 12

p q

x y

⇒ y = `(x)

(14)

⇒ q = L(p)

(15)

Consider sectors I and IV in Figure 3. The original Lorenz and its Ginitransformed curves both have critical (circle) and discriminant (square) points. On the transformed curve the critical point C 00 (xc , yc ) is such that `0 (xc ) = 0 with xc = arg max `(x), as on the original curve the critical point C(pc , qc ) is such that L0 (pc ) = 1 with pd = arg max |L(pw ) − pw |. On the transformed curve the discriminant point is at the intersection of the curve `(x) with the vertical line x = 1, i.e. xd = 1 such that yd = `(1), in the same way pd is such that qd = L(pd ) = 1 − pd . Moreover, qd = 1 − pd ⇒ 0.5(xd − yd ) = 1 − 0.5(xd + yd ) ⇒ xd = 1 ⇒ yd = `(1)

(16)

shows that, not depending on the original Lorenz curve, the Gini-transformed Lorenz curve y = `(x) always intersects the line x = 1 at D00 (xd = 1, yd = `(1)) which is equivalent to D(pd , qd = L(pd )) on the Lorenz curve at the intersection of qw = L(pw ) with q = 1 − p which reads as F(w) + L(pw ) = 1.

3.2

The Zanardi asymmetry index

This section concerns the derivation of Zanardi asymmetry index. As an example, data have been extrapolated from the original Lorenz 1905 article [14] on the income distribution in Prussia in 1901.6 6 By extrapolating data from the Lorenz 1905 paper, a dense L-curve has been interpolated by means of a Piecewise Cubic Hermite Interpolating Polynomial routine implemented in MATLABTM [15]. The associated `-curve has then been obtained as described in Section 3.1.

I: the L-curve q=L(p)

II: Reflection

-0.2

0.6

-0.4

M0 C

0.4

-q

q=L(p)

0.8

-0.6

0.2

-0.8

O 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-1

0.1

0.2

0.3

0.4

0.5

0.6

IV: Stretching -> the Gini transformed L-curve y=l(x)

III: Rotation

0.7

0.8

0.9

0.4

y=p-q

0.2

Ar=K dG r

C ''

Ap=K d Gp

0.3

D ''

0.15

0.2

0.1

0.05

M'' 0

O '' 0

0.2

0.4

0.6

0.8

E '' 1.2

1.4

1.6

1.8

0 0

0.2

0.4

0.6

0.8

1.2

1.4

x=p+q

Figure 3: The Figure 1 L-curve on the plane P ≡ (p, q) and its Gini transformed `-curve on the P 00 ≡ (x, y) plane. The new discriminant point D00 is indicated by the square, the new critical point C 00 is indicated by the circle. The area A under the `-curve is equivalent to the overall concentration G. The two areas Ap and Ar allows for estimating specific concentrations G p and G r in the poor and rich side respectively. The discriminant point separates the poor consumers side from the rich one. Figure 1 and sector I in Figure 3 show that about pd = 65% of consumers are poor and they accumulate about qd = 35% of total income, see Table 1. This suggests there is a certain degree of inequality in income distribution since it means that if there were 100 people and 100 dollars, 65 people share 35 dollars, i.e. Rdp = 0.54 cents on average, while the other 35 people share the remaining 65 dollars, i.e. Rdr = 1.86 dollars on average: therefore, the average per-capita rich income is Od = 3.44 times the average per-capita poor one. Moreover, being pc = 0.8140 > pd it follows that asymmetry is positive in disadvantage of the poor side, which should turn into being more homogeneously poor than what the rich side is heterogeneously rich: the poor side is therefore less concentrated than the rich one. To meet the Zanardi index as presented in [25, 26] define the factor Kd =

pd qd 2

(17)

which, according to previous estimates, gives Kd = 0.1140. Notice that sector IV in Figure 3 shows two areas. For x ∈ [0; 1] the area Ap is limited by the arc 00 D 00 and the segment O 00 D 00 , for x ∈ [1; 2] the area Ar is limited by the arc ^ O 00 E 00 and the segment D 00 E 00 . Estimators of such areas are ^ D ( Rx Ap = 0 d `(x)dx − y2d R2 where yd = l(xd ) : xd = 1 (18) Ar = xd `(x)dx − y2d 13

By numeric integration of the `-curve it has been estimated that Ap = 0.0281 < Ar = 0.1021. According to Zanardi [27, 28], the ratios Gp =

Ap Kd

, Gr =

Ar Kd

(19)

define the Gini indexes of the poor and rich sides respectively: sector IV in Figure 3 and Table 1 show that G p < G r , which means the rich class is G r /G p = 3.63 times more concentrated than the poor one. The Zanardi asymmetry index is defined as follows Zd = 2Kd

Ar − Ap 1 − A(1) Gr − Gp =2 =4 ∈ [−1; +1] G G A(2)

(20)

where A(x) is the area below `(x) in the interval [0; x]. Therefore, Zd depends on the discriminant point D(pd , L(pd )) = Γ−1 [D00 (1, `(1))], not on the critical one, but the two are related: it has been observed that the direction of the asymmetry of the Lorenz curve depends upon pd ≷ pc , hence the critical point should play some role. If G p < G r , as in the example of Figure 3 and 4, then Zd > 0 means the asymmetry profile is positive: the rich group is more heterogeneous than the poor one. This is of course a negative aspect for the poor side: a lot of poor consumers almost equally and gradually share a few of total income while a restricted set of rich consumers differently share the hugest part of total income. In the present example it can be measured that the overall concentration G = 0.4293 is quite high and that Zd = 0.3445 evaluates an almost high degree of inequality in Prussia in 1901. If Zd < 0 there is a negative asymmetry. If Zd = 0 the distribution is perfectly symmetric, that is the rich and poor groups are (un-)equal the same: note that Zd = 0 does not mean income is equally distributed, it only means the Lorenz curve is perfectly symmetric and that the poor and rich sides are within (un-)equal the same, only in such a case everything is ruled by G alone. It is then worth mentioning that, by definition, the Zd index is normalized on the overall G index, therefore it can be suitably involved in direct comparison of any two or more sampling Lorenz curves taking care of concentration, heterogeneity and asymmetry, that is inequality, even in the case of intersecting curves. The Zd index evaluates asymmetry by means of heterogeneity, while splitting the sample into rich and poor sub-samples, and concentration, while considering their specific concentrations and the overall one. As a matter of fact, the Zd index gives a more reliable measure of inequality than G because it evaluates the income class-divide δ = G r − G p between poor and rich sub-samples normalized by the overall concentration G. Therefore, the Zanardi index provides all what required by the inequality Definition 1.

3.3

Characteristic points

The previously discussed properties ensure that characteristic points D and C uniquely identify a L-curve such as D00 and C 00 uniquely identify a `-curve: characteristic points introduce uniqueness. Indeed, any Γ-transformed L-curve ` intersects the vertical line x = 1 at one and only one discriminant point D00 and it has one and only one critical point C 00 at the maximum distance form the x-axes. But two7 , or even more, curves `j may share the same D00 with different points Cj00 or they may share the same C 00 with different points Dj00 while having the same asymmetry profile: knowing just one of the two characteristic points is not sufficient to uniquely identify a specific ` curve in terms of Zd , just like the overall Gini index G is not sufficient to identify a L-curve and draw conclusions on inequality. On the contrary, through the couple of characteristic points D00 and C 00 may pass one and only one curve. Therefore, either one knows `, and then uniquely finds D00 to evaluate Zd , otherwise she needs to estimate ` to obtain both C 00 and D00 to evaluate Zd .8 A desirable property required to an inequality index is its uniqueness: which means it would be nice developing an inequality index whose estimate cannot be common to two or more different distributions. Even though it is a hard task developing such a kind of index, because it is an overall synthetic statistics referred to a distribution as a function, that of uniqueness is an important topic mainly for applications. By exploiting the definition of the Zanardi asymmetry as an inequality index the uniqueness property can be partially approached. The following reasoning applies to the case of Zd > 0 represented in Figure 4, in the opposite case a perfectly equivalent reasoning can be developed mutatis mutandis. 00 E 00 and the segment D 00 E 00 . ^ In Figure 4, the rich side is limited by the arc D Hence, the area Ar , which allows for evaluating G r , can be partitioned into two sectors: Awr for weakly-rich consumers and Asr for strongly-rich ones. Awr is 00 C 00 and by the two segments I 00 D 00 ^ the area of the sector identified by the arc D sr 00 00 and I C . On the other hand, A is the area of the sector identified by the 00 E 00 and by the two segments I 00 C 00 and I 00 E 00 . Both the critical C 00 and ^ arc C the discriminant D00 points come into play. As it can be seen Ar = Awr +Asr . Therefore αwr = Awr /Ar and αsr = Asr /Ar can be conceived as importance weights of the two distinct rich sub-groups in determining the imbalance measure G r : notice that this further splitting depends on the critical point C 00 . In the present example, a by eye inspection is sufficient to conclude that αsr > αwr , therefore most of the responsibility in the positive asymmetry which disadvantages the poor side is mainly due to the strongly-rich consumers against the weakly-rich ones. 7 See

Figure 2 as a reference in the case of L-curves: the same applies to their `-curves. literature there are several proposed methods to estimate an approximation of the Lorenz curve on real data, any tentative of listing them would be too restrictive. Just to mention two of the most cited ones see [9] and [12]. In this study the involved method is a polynomial interpolation as reported in footnote 6. 8 In

The Gini transformed L-curve y=l(x) 0.35

C ''

0.3

D ''

y=pw-q w

0.25

I '' 0.2

0.15

0.1

0.05

M'' 0

O ''

0.2

0.4

0.6

0.8

J'' 1

1.2

E '' 1.4

1.6

1.8

x=pw+q w

Figure 4: The Gini transform of the L-curve and characteristic points. The problem is then to evaluate such areas. To this end the service point I 00 is considered. I 00 lies on the negatively-sloped9 the straight-line y = a + bx hosting D00 E 00 and it coincides with the intersection point with the vertical line x = xc hosting J 00 C 00 : its abscissa is therefore xi ≡ xc . To determine yi it takes knowing the equation of the straight line y = a + bx passing through D00 and E 00 . Being D00 (1, yd ) and E 00 (2, 0) by definition, it then follows that the equation of the straight-line is (y − yd )/(ye − yd ) = (x − xd )/(xe − xd ), hence y = yd (2 − x).10 By setting x = xc it follows that yi = yd (2 − xc ), which involves coordinates of both the critical and discriminant points and states a relationship between coordinates of the characteristic points, which turn into being related with each other whatever the Lorenz curve is.11 As the characteristic points are unique for a curve, I 00 (xc , yd (2 − xc )) is unique as well. Therefore, the weakly-rich area is evaluated as Z xc (yi + yd )(xc − 1) wr A = `(x)dx − (21) 2 1 which subtracts the area of the trapeze M000 J 00 I 00 D00 from the area below `(x) is because asymmetry is positive, in case Zd < 0 then I 00 would lie on the positivelysloped straight-line hosting O00 D00 . 10 If Z < 0 then the straight-line passing through O 00 and D 00 is (y − y )/(y − y ) = o o d d (x − xo )/(xd − xo ), which gives y = yd x. 11 In case Z < 0 then y = y x still depends on the characteristic points. i d d c 9 This

00 C 00 . The strongly-rich area is ^ under the arc D Z 2 (2 − xc )yi Asr = `(x)dx − 2 xc

(22)

which subtracts the area of the triangle I 00 J 00 E 00 from the area below `(x) under 00 E 00 . It then follows that ^ the arc C Ar

= Awr + Asr Z 2 (yi + yd )(xc − 1) (2 − xc )yi = `(x)dx − + 2 2 1 where xi = xc , yi = yd (2 − xc )

(23)

can then be involved into (19) to evaluate G r .12 Of course, the numeric value of Ar is the same for both cases, indeed (23) simplifies into (19), but this second expression makes it completely dependent on the unique characteristic points of the curve.

3.4

The LGZ methodology to measure inequality

Before proceeding further some results are worth summarizing about what is here labeled as the LGZ methodology: which stands for the Zanardi asymmetry measured on the Gini-transformed Lorenz curve of the income distribution. As such, the described methodology to the measurement of inequality proposes to transform the income distribution F on the (w, p)-plane into its equivalent Lorenz curve L on the (p, q)-plane to obtain its Gini transform ` on the (x, y)plane: F, L and ` are equivalent representations of income W . The proposed methodology allows for detecting how many consumers are owning what share of total income, while conventionally distinguishing the poor side from the rich one, this is what the discriminant point D(pd , L(pd )) 7→ D00 (1, `(1)) does. The critical point C(pc , qc ) 7→ C 00 (xc , yc ) allows for detecting the direction of the imbalance in the distribution. These characteristic points are related with each other and uniquely associate with a given curve L 7→ `: two curves sharing these points are necessarily the same curve. By separating the poor form the rich side on `, the Zanardi asymmetry index Zd compares the class-divide δ = G r − G p to the overall concentration G, which allows for detecting a measure of asymmetry as a measure of inequality for comparisons, even when the concentration curves intersect with each other, this is not possible by means of the Gini index alone. Moreover, not depending on the direction of asymmetry, dividing Aj = G j Kd (j = p, r) by A it is possible weighting the relevance of each side in generating the imbalance of the distribution. Further than this, depending on the direction 12 If

Zd < 0 there would be a weakly-poor and a strongly-poor sector, respectively under 00 D 00 and O 00 C 00 . Hence, with the same reasoning, but mutatis mutandis, the area ^ the arcs C^ to measure is Ap , hence (19) will give G p . Therefore, instead of αwr and αsr one will find αwp and αsp .

statistics pd qd pc qc Kd Rdp Rdr Od A=G Ap Ar Gp Gr δ Zd

Prussia 1892 0.6380 0.3613 0.8140 0.5157 0.1153 0.5663 1.7644 3.1156 0.3933 0.0160 0.1006 0.1387 0.8728 0.7341 0.4303

Prussia 1901 0.6500 0.3509 0.8140 0.4957 0.1140 0.5398 1.8547 3.4359 0.4293 0.0281 0.1021 0.2466 0.8950 0.6484 0.3445

∆1901 1892 +0.0120 -0.0104 0.0000 -0.0200 -0.0013 -0.0265 +0.0903 +0.3203 +0.0360 +0.0121 +0.0015 +0.1079 +0.0222 -0.0857 -0.0858

Table 1: LGZ inequality measurement method statistics on data extrapolated from Lorenz 1905 [14]: income distribution in Prussia in 1892 and 1901. of asymmetry, it is also possible weighting the relevance of specific sub-classes in determining the detected inequality, that is αwj and αsj . Therefore, the LGZ methodology does all what required by the Definition 1 of inequality by adding some further details: the LGZ method has been also applied to data extrapolated from the Lorenz 1905 article as regarding the situation in Prussia in 1892 for comparison with 1901, see Table 1. As the illustrations proposed by Lorenz show, income is more concentrated in 1901 than in 1892. Lorenz himself observes this without knowing how to compute the Gini index: G1892 = 0.3933 → G1901 = 0.4293. Therefore it all seems to suggest an increase of inequality. But Lorenz claims: The curves may not always give so clear an answer as in the previous illustrations, because opposing tendencies may exists at the same time, but the diagram will always tell what has happened. Hence, one can think Lorenz himself was to some extent aware that his method can only measure concentration while to give insights on inequality some further details are needed. By applying the LGZ method it seems that such opposite and hidden tendencies can be detected: see Table 1. First of all it can be observed the discriminant point has moved with a +0.0120% in poor consumers with −0.0104% of income to share: it slided down to the right, i.e. rolling away from M0 toward E on the (p, q)-plane or rising up from M000 along the vertical line x = 1 on the (x, y)plane. Therefore, ∆Rdp = −0.0265 and ∆Rdr = +0.0903 lead to ∆Od = +0.3203: more than in 1892, in 1901 it takes more per-capita poor incomes to balance with a per-capita rich one. Everything sustains the hypothesis of an increase of inequality, as ∆G = +0.0360 would confirm: it is worth noticing that Rdp , Rdr 18

and Od are average estimates. Also, notice that the imbalance profile has not changed, indeed the critical point is always above the discriminant one (i.e. pc > pd in both years) but its distance from the equi-distribution line increased (i.e. qc1892 > qc1901 ). The changes in the specific concentrations are positive ∆G p = +0.1079 and ∆G r = +0.0222: the rich side is always more concentrated than the poor one. Therefore, the imbalance in income distribution is always disadvantaging the poor side, but since ∆G p > ∆G r then the poor side becomes less disadvantaged, which does not mean it becomes advantaged as for the rich side. Moreover, the class-divide decreased by ∆δ = −0.0857, opposite to the increase in the overall concentration ∆G = +0.0360. These may be those opposite tendencies Lorenz was referring to. Indeed, it can be observed that ∆Zd = −0.0858 to mean that inequality has decreased because, even though concentration has increased the class-divide has decreased more: the disadvantage the poor consumers owe to the income distribution in 1901 is lower than it was in 1892. All such opposite tendencies are hidden in the Lorenz curve and cannot be grasped by the Gini index: consistently with Definition 1, the LGZ method suggests the Zanardi asymmetry as a reliable measure of inequality.

Asymmetry and heterogeneity, as if they were important

Without distinguishing the marginal propensity to consumption13 for durable and not-durable goods or other criteria, there is evidence that the MPC (c0 ) associated to the minimum income level (wmin ) is greater than that MPC (c1 ) associated to the maximum (wmax ) one, i.e. 1 > c0 c1 > 0. Moreover, as regarding to a given sample of consumers, there is evidence that the MPC speed of decay slows down as the income level increases: the more a consumer gets richer the more she saves therefore, at least beyond some certain income level, the more w → wmax the less the MPC decreases to become almost flat and constant at the top incomes. This phenomenology is widely diffused as a regularity, and it couples with the inequality null hypothesis of Definition 1: in a sense, the two are different sides of the same coin, and, beyond Proposition 1, they both relate to the Keynesian multiplier of income as stated in Thesis 1. To develop a multiplier which takes care of inequality, an inequality-adjusted MPC estimator is needed. To this end consider (w) 7→ n(w) =

w − wmin ≡ nw ∈ [0; 1] wmax − wmin

(24)

hence nw ∈ [0; 1] is equivalent to w ∈ [wmin ; wmax ]. Moreover, according to the inequality null hypothesis, in the standard case the higher the positive asymmetry of the L-curve the higher the inequality of income distribution, with disadvantage of the poor side, see Section 3. It can then be 13 MCP

from here on, whose numeric values are represented with c’s.

assumed that, at any given inequality degree, the MPC decays steeply at the lower levels of income to change almost flatly at the top incomes. Therefore, given two income distributions with different but positive inequality degree (see Table 1 as an example), as the income levels grow the MPC decays faster on the income distribution with higher inequality. As a measure of inequality the Zanardi asymmetry index (20) defines (Zd ) 7→ ζ(Zd ) =

1 + Zd ≡ ζd ∈ [0; 1] 2

(25)

being this a monotonic transformation it does not affect the described phenomenology which relates MPC with inequality: the more Zd → 1− the more ζd → 1− is advantaging the rich side against the poor one, while the more Zd → −1+ the less ζd → 0+ is disadvantaging the poor side. If Zd = 0 then ζd = 0.5 the two sides are concentrated the same. A simple MPC-curve obeying the above regularities is14 θ(nw |ζd ) = c0 + (c1 − c0 )nw eζd (1−nw ) ∈ [c0 ; c1 ], ∀ζd ∈ [0; 1]

(26)

(27)

because 0 < c1 c0 < 1 and nw ζd < 1, therefore: as regarding to a given distribution, the higher is nw the lower the MPC at nw on the MPC-curve (26). Moreover, it can be also checked that ∂θ(nw |ζd ) = (c1 − c0 )(1 − nw )eζd (1−nw ) < 0 , ∀nw ∈ [0; 1] ∂ζd

(28)

Therefore, given two income distributions F 1 and F 2 , where the implied Lcurves are such that ζd1 < ζd2 , it then follows that θ(nw |ζd2 ) decays faster than θ(nw |ζd1 ) at any nw . Accordingly, the more inequality is negative (or less positive) the less the income distribution disadvantages the poor consumers, hence the higher the MPC at any nw = n(w). As matter of fact, the higher the MPC the higher the multiplier of income: therefore, this modeling extends Proposition 1 and provides the basis of Thesis 1. By substituting for (24), the MPC-curve (26) admits an equivalent specification in terms of w, say θ̂(w|ζd ). Since w = F −1 (pw ) = F1−1 (µqw ) = F1−1 (µL(pw )), both θ(nw |ζd ) and θ̂(w|ζd ) describe the MPC-curve at every quantile of the distribution F of the L-curve (see Figure 5), that is along 14 Of course, there may be given also other formulations of the MPC-curve while obeying the above phenomenology, just like there may be given several welfare functions satisfying essential properties, see [5] and [3]. The here proposed expression is simple and general enough to match with commonly recognized properties of the MPC.

F. Once F is known then everything can be analytically explained, indeed θ̂(w|ζd ) ≡ θ̂(F −1 (pw )|ζd ) ≡ θ̂(F1−1 (µqw )|ζd ) ≡ θ̂(F1−1 (µL(pw ))|ζd ). According to the MPC-curve in (26), average MPC estimates on the poor [0; nd ] and rich [nd ; 1] sides can be obtained by evaluating θ(•|ζd ) at sample points as representatives of the two sides. By intuitively choosing side-specific middle points as representatives, the estimators of the side-specific MPC can be defined as15 p p θ (nd |ζd ) = θ n2d |ζd ≡ c r (29) 1+nd r θ (nd |ζd ) = θ 2 |ζd ≡ c while θ(nd |ζd ) = cd

(30)

is the value of the MPC at the discriminant point nd = n(wd ), where wd = F −1 (pd ). It can then be checked that 0 < cp < cd < cr < 1

(31)

is always fulfilled for any value of asymmetry ζd = ζ(Zd ), not depending on the underlying income distribution. The standard MPC involved in the Keynesian multiplier is an overall (i.e. aggregate) measure referred to an income distribution as a whole, without considering heterogeneity, nor concentration and asymmetry: i.e. without considering inequality. Therefore, to improve the usual estimator, at first some degree of heterogeneity is introduced by evaluating the average MPC as a weighted average of rich and poor propensities on the MPC-curve, with weights given by the share of rich (1 − pd ) and poor (pd ) consumers determined by means of the LGZ method: asymmetry and concentration are consequently introduced. Therefore, a simple average MPC estimator can be the following c(nd |ζd )

(32)

Expressions (32) involve both the MCP-curve, with side-specific estimators, and the LGZ method. It is an estimator in the sense that as the sampling distribution of income changes the Lorenz curve changes accordingly, therefore to any given sample of income W from a distribution F a Lorenz L and a MPC curve θ are associated. Equation (32) defines an estimator of the average MPC as a function of the normalized threshold nd = n(wd ), equivalent to the discriminant income wd = F −1 (pd ) identified by the discriminant point D(pd , qd ) on the Lorenz curve associated to the income distribution. Once the income distribution is known, the LGZ method provides the implied Lorenz curve with its critical and discriminant points, needed to evaluate the asymmetry and income threshold which conventionally separates poor and rich incomes. 15 Of

course, there may be other ways to estimate cp and cr , the proposed ones seem to be almost intuitive.

Lorenz curve 1

.7 (1+n d )/2

.6 .5 .4

n: normalized income levels

MPC curve 1

.2 .1

.7 .6 .5 .4 .3 .2

n d /2 d cp c

0 1 .9 .8 .7 .6 .5 .4 .3 .2 .1 0

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

θ(n|ζd ): MPC

p: cumulative share of consumers

Figure 5: A representative example of the MPC along the Lorenz curve for Prussia 1901: c0 = 0.95 and c1 = 0.35 have been conventionally set. Estimates of cp , cd and cr are identified on the MPC-curve θ(n|ζd ) at nd /2, nd and (1 + nd )/2 respectively. Such values are also represented on the Lorenz curve. The first vertical line in the MPC-curve panel represents c0 , the last one represents c1 . Therefore, once an income distribution is given, the LGZ method and the MPCcurve allow to estimate of the average MPC for the whole economy as c = cp pd + cr (1 − pd ) = cr + (cp − cr )pd

(33)

obtained by substituting (29-30-31) into (32): numeric values for the present example are reported in Table 2. Figure 5 maps the L-curve onto the MPC-curve, which is bounded within c0 and c1 . The value 1 − c0 represents the saving propensity for the poorest consumers, which is almost zero, while 1 − c1 is the saving propensity for the richest ones, which is significantly different from zero. The discriminant point D(pd , nd ) is mapped onto the MPC-curve so splitting it into two parts: θp (nw |ζd ) = θ(nw ≤ nd |ζd ) for poor consumers and θr (nw |ζd ) = θ(nw > nd |ζd ) for rich ones. As it can be seen, in the poor side the MPC decays fast within a restricted set of values for nw , i.e. w, while in the rich part it decays slowly within a wider set of values for nw .

MPC G Zd ζd c0 cp cd cr c1 c

Prussia 1892 0.3933 0.4303 0.7172 0.9500 0.7553 0.6077 0.4638 0.3500 0.6400

Prussia 1901 0.4293 0.3445 0.6723 0.9500 0.7668 0.6243 0.4459 0.3500 0.6545

∆1901 1892 +0.0360 -0.0858 -0.0449 0.0000 +0.0015 +0.0166 +0.0091 0.0000 +0.0145

Table 2: MPC estimates on the distribution of income consistent with quantiles of the Lorenz curve for Prussia in 1892 and 1901, [14]. c0 and c1 are the upper and lower bounds, cp is the average MPC on the poor side, cd is the MPC corresponding to the income at the discriminant point, cr is the average MPC on the rich side. c is the overall average MPC. Table 2 refers to estimates based on data extrapolated from the quantiles of the Lorenz curve obtained by means of the LGZ method as described in Section 3.4. The minimum and maximum levels of MPC has been conventionally set and maintained constant in both years: their magnitude does not affect the main result, that is as far as inequality decreases by ∆Zd = −0.0858 the average MPC increases by ∆c = +0.0145, even though the overall concentration increases by ∆G = +0.0360. Therefore, an increase of concentration does not necessarily imply an increase of inequality and a decrease of the MPC. Mainly, it depends how the configuration of the income distribution changes, that is depending on what side owes the greater advantage of the change. As reported in Tables 1, the overall concentration increased but the class-divide decreased more reducing inequality: that is, the poor side owed to the income distribution change a lower disadvantage. Accordingly, the increase in the range of opportunity of enjoyment has improved more within the poor side than within the already wide in the rich side. On the other hand, if a poor gets more income she will spend it (almost) all, while a rich consumer would not modify her propensity so evidently.

The inequality multiplier

The textbook income multiplier story has been told by Proposition 1: if the average MPC increases then the multiplier increases because the propensity to save decreases and, therefore, GDP grows. Differently said, due to an increase of consumption opportunities (demand) production (supply) is stimulated. As suggested in Thesis 1, this section aims at showing why and how inequality matters for the income multiplier. The MPC-curve θ(n|ζd ) has been defined in (26) depending on the Lorenz curve

as characterized by a given degree of inequality. Therefore, the inequality index ζd = ζ(Zd ) in (25) comes then into play with all its terms: i.e. heterogeneity, concentration and asymmetry. Hence, cp and cr in (29) are estimates of the class-specific propensities. Moreover, the average MPC estimator has been defined in (32-33). On this basis, to develop an inequality adjusted multiplier of income (shortly the inequality multiplier ) it is worth considering the following equivalences nw =

w − wmin − wmin

wmax

⇔ w = wmin + (wmax − wmin )nw

F −1 (pw ) − wmin wmax − wmin F −1 (L−1 (qw )) − wmin qw = L(pw ) ⇔ nw = wmax − wmin pw = F(w) ⇔ nw =

(34)

which imply nd =

F −1 (pd ) − wmin F −1 (L−1 (qd )) − wmin = ≡ η(pd ) wmax − wmin wmax − wmin

(35)

Therefore, nd = n(wd ) = η(pd ) and the average MPC estimator reads as c(nd |ζd ) = θr (η(pd )|ζd ) + [θp (η(pd )|ζd ) − θr (η(pd )|ζd )]pd ≡ χ(pd |ζd ) > 0 (36) which is a function of the discriminant point conditioned by the asymmetry of the Lorenz curve implied by the underlying income distribution. As such, c(nd |ζd ) = χ(pd |ζd ) takes care of heterogeneity, concentration and asymmetry, i.e. of inequality. By implementing (36) into the simplest definition of the multiplier, it then is adjusted for inequality16 λ(pd |ζd )

1 1 − χ(pd |ζd )

(37)

Therefore, the inequality adjusted multiplier depends on the average MPC adjusted for inequality. This propensity depends on the class-specific shares of consumers and income. The class-specific propensities are found by means of the MPC-curve at side-specific sample points of the Lorenz curve (see Figure 5), and they also depend on the asymmetry. The discriminant point and the asymmetry depend on the Lorenz curve associated to the income distribution. Therefore, in the end, the income multiplier depends on the income distribution. Hence, by means of the inequality adjusted MPC the income distribution enters the macroeconomics of the Keynesian multiplier while taking care of heterogeneity, concentration and asymmetry all at once, which is what Thesis 1 suggests. Of course, as an estimator it returns an estimate, i.e. a number, but in so doing 16 Clearly, this expression refers to the basic multiplier expression, i.e. for a closed economy without Government, as in Proposition 1.

MPC G δ Zd ζd c(nd |ζd ) = χ(pd |ζd ) λ(pd |ζd )

Prussia 1892 0.3933 0.7341 0.4303 0.7172 0.6400 2.7777

Prussia 1901 0.4293 0.6484 0.3445 0.6723 0.6545 2.8942

∆1901 1892 +0.0360 -0.0857 -0.0858 -0.0449 +0.0145 +0.1165

Table 3: The inequality adjusted multiplier of income for Prussia in 1892 and 1901, [14]. it embeds all the needed properties to link its value to the inequality profile implied by the income distribution F as described by the implied Lorenz curve L. Differently said, since the discriminant point D(pd , qd ) is such that pd = 1−qd = 1 − L(pd ) = F(wd ), the multiplier depends upon q = L(p) on (p, q)-plane which is equivalent to p = F(w) on the (w, p)-plane: this means the inequality adjusted multiplier embeds heterogeneity, concentration and asymmetry of the income distribution. Moreover, since Zd = 2Kd δ/G, where δ = G r − G p and Kd = pd qd /2, it embeds the inequality of the income distribution as estimated on the implied Lorenz curve by means of the LGZ method. Therefore, by considering heterogeneity, concentration and asymmetry at once, the proposed multiplier is said to be adjusted for inequality because it depends on the inequality adjusted MPC. Thesis 1 states that if inequality increases then the multiplier decreases, that is: the relationship between inequality ζd = ζ(Zd ) and the inequality multiplier λ(pd |ζd ) is negative. This can be checked by computing derivatives of (37) to account for inequality increasing effects. Therefore, the sufficient condition for Thesis 1 to be fulfilled is ∂χ(pd |ζd ) ∂χ(pd |ζd ) ∂λ(pd |ζd ) −2 = (1 − χ(pd |ζd )) < 0 ⇔ <0 ∂ζd ∂ζd ∂ζd According to (36) this condition reads as r ∂θr (η(pd )|ζd ) ∂θ (η(pd )|ζd ) ∂θp (η(pd )|ζd ) < − pd ∂ζd ∂ζd ∂ζd

(38)

(39)

which simplifies into (1 − pd )

∂θp (η(pd )|ζd ) ∂θr (η(pd )|ζd ) <− pd ∂ζd ∂ζd r

(40) p

d )|ζd ) d )|ζd ) < 0 and ∂θ (η(p < By observing that pd > 0 and (1−pd ) > 0 while ∂θ (η(p ∂ζd ∂ζd 0 it then follows that the l.h.s. of (40) is always negative while the r.h.s. is always positive: therefore (40) is always fulfilled, and this satisfies the sufficient

condition (38) for Thesis 1. As a further remark notice that this outcome is consistent with Definition 1, improves Proposition 1 and explains Thesis 1. Therefore, if inequality decreases then the MPC increases and, as a consequence, the multiplier increases as well. As Table 3 shows, the multiplier may increase even if the concentration increases: if the increase in concentration is more than compensated by a decrease in the class-divide, then inequality decreases and stimulates the propensity to consumption.

Concluding remarks

Recently the Great Recession, the increase in economic inequality and austerity policies have brought the attention of economists to the issues of redistribution of income and the relationship between growth and inequality. Within this debate the multiplier has taken on a central role: what is its value has been discussed in many papers. But an analytical formulation of the multiplier adjusted for inequality did not exist up to now. This note fills this gap. As a detail, it is worth stressing that no free parameter has been involved. The proposed modeling is analytic and phenomenologically based on empiric evidence. The key parameter involved is the Zd inequality index, which is endogenous to the LGZ methodology and receives a theoretical interpretation in measuring the impact of inequality on the multiplier. The main finding is that when inequality increases the multiplier decreases, (Thesis 1): this has been shown by applying a proposed inequality adjusted multiplier to the case of the income distribution in Prussia in 1892 and 1901, where data have been extrapolated from the original 1905 paper of Lorenz [14]. This formula presents a negative relationship between inequality and multiplier value. Ceteris paribus, the same amount of spending - it can be argued - will reduce GDP whenever inequality increases, i.e. inequality hampers growth.

Acknowledgements The authors thank Bruce C. N. Greenwald for suggestions and comments. The paper has been jointly developed within the research program of the Economies in transition unit at I.R.E.S. Piemonte, the MatheMACS project at DiSES, and the Macroeconomic Efficiency and Stability INET task-force project at Columbia University. The authorsâ&#x20AC;&#x2122; views and opinions are their own and do not involve the responsibility of their institutions. The authors declare no conflict of interest.

References [1] Atkinson, A.B., (1972), On the measurement of inequality, Journal of Economic Theory, 2, pp. 244-263. 26

[2] Atkinson, A.B. and Brandolini, A., (2001), Promise and pitfalls in the use of ”secondary” data-sets: Income inequality in OECD countries as a case study, Journal of Economic Literature, 39(3), pp. 771-779. [3] Atkinson, A.B. and Brandolini, A., (2015), Unveiling the ethics behind inequality measurement: Dalton’s contribution to Economics, The Economic Journal, 125(583), pp. 209-220. [4] Blanchard, O. and Leigh, D. (2013), Growth forecast errors and fiscal multipliers, IMF Working Paper No. 13/1 (Washington: International Monetary Fund). [5] Dalton, H., (1920), The measurement of the inequality of incomes, Economic Journal 30(119), pp. 348-361. [6] Damgaard, C. and Weiner, J., (2000), Describing inequality in plant size or fecundity, Ecology, 81(4), pp. 1139-1142. [7] Frosini, B.V., (2012) Approximations and decompositions of Gini, PietraRicci and Theil inequality measures, Empirical Economics, 43(1), pp. 175197. [8] Gastwirth, J.L., (1971), A general definition of the Lorenz curve, Econometrica, 39(6), pp. 1037-1039. [9] Gastwirth, J.L., (1972), The Estimation of the Lorenz Curve and Gini Index, The Review of Economics and Statistics, 54(3), pp. 306-316. [10] Gini, C., (1912), Variabilità e Mutuabilità. Contributo allo studio delle distribuzioni e delle relazioni statistiche. C. Cuppini, Bologna. [11] Gini, C., (1932), Intorno alle curve di concentrazione, Metron, 9, pp. 2-76. [12] Kakwani, N. C. and Podder, N., (1976), Efficient estimation of the Lorenz curve and associated inequality measures from grouped observations, Econometrica, 44(1), pp. 137-148. [13] Lindert, P., (2004), Growing public: social spending and economic growth since the Eighteenth Century. Vol. 1. Cambridge University Press. [14] Lorenz, O.M., (1905), Methods of measuring the concentration of wealth. Publications of the American Statistical Association, 9(70), pp. 209-219. [15] MATLAB, 8.4.0.150421 (R2014b), The MathWorks Inc., Natick, Massachusetts. [16] Merton, R.K., (1968), The Matthew effect in science. Science, 159(3812), pp. 1-8. [17] Milanovic, B., (2002), True World income distribution, 1988 and 1993: First calculations, based on household surveys alone, The Economic Journal, 112(476), pp. 51-92. 27

[18] Pietra, G., (1915), Delle relazioni tra gli indici di variabilità Nota I. Atti del Regio Istituto Veneto di Scienze 74: pp. 775-792. [19] Pigou,A. (2012) Wealth and Welfare, Macmillan. [20] Piketty, T., (2014) Capital in the 21st Century. Harvard Unversity Press. [21] Ricci, U., (1916), L’indice di variabilità e la curva dei redditi. Giornale degli Economisti e Rivista di Statistica Serie Terza, 53: pp. 177-228. [22] Rothschild, M. and Stiglitz, J.E., (1971), Increasing risk: I, a definition., Journal of Economic Theory, 2(3), pp.225-243. [23] Stiglitz, J. E., The price of inequality. How today’s divided society endangers our future, W.W. Norton & Company. [24] Taguchi, T., (1968), Concentration-curve methods and structures of skew populations, Annals of the Institute of Statistical Mathematics, 20(1), pp. 107-141. [25] Tarsitano, A., (1987), L’interpolazione della spezzata di Lorenz, Statistica, 43(3), pp. 437-451. [26] Tarsitano, A., Measuring asymmetry of the Lorenz curve, Ricerche Economiche, 43(3), pp. 507-519. [27] Zanardi, G., (1964), Dell’asimmetria condizionata delle curve di concentrazione. Lo scentramento. Rivista Italiana di Economia, Demografie e Statistica, 18, pp. 431-466. [28] Zanardi, G., (1965), L’asimmetria statistica delle curve di concentrazione. Ricerche Economiche, 19, pp. 355-396.