Lionel Schouveiler

Passive stability control of hanging articulated pipes using an additional mass

Keywords: pipe, futter, divergence, passive control

Depending on their physical characteristics, freely hanging articulated pipes can undergo divergence-or flutter-type instability above a critical velocity of the flow they conveye. The effect of adding a point mass, and more precisely influence of the mass magnitude and location, on the stability of such pipes is investigated by means of a linear analysis. It is found that this additional load can have a stabilizing or destabilizing effect depending on its location along the pipe and that the effect is all the more important as the mass is high. The influence of the mass on the shape of the destabilized linear mode is also discussed.

Introduction

Pipes conveying fluid are encountered in many industrial applications and their dynamics and stability have motivated numerous studies. A freely hanging pipe is one of many configurations of pipes conveying fluid, it consists of a vertical pipe with a fixed upstream end and a free downstream end where 5 the fluid discharges to the atmosphere. When the velocity of the conveyed flow exceeds a critical value, an hanging pipe initially straight can be destabilized and either static or oscillatory deflection occur, respectively referred to as divergence or flutter. One of the first comprehensive study on the stability of hanging pipes was devoted to articulated pipes that is pipes made of rigid seg-10 ments connected by flexible joints. In this theoretical and experimental work, Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF][START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. II. Experiments[END_REF] showed that, depending on their physical characteristics, these pipes can be destabilized by divergence or by flutter. In contrast, a study by Païdoussis [START_REF] Païdoussis | Dynamics of tubular cantilevers conveying fluid[END_REF] revealed that hanging continuously flexible pipes are only subject to planar flutter.

The effect of adding a mass at the free end of continuously flexible pipes has been considered by Copeland and Moon [START_REF] Copeland | Chaotic flow-induced vibration of a flexible tube with end mass[END_REF]. Their experiments revealed that, depending of the end mass value, the flexible pipe can exhibit various routes to chaos as the flow velocity is increased, and a great variety of dynamical states. This richness of states, including periodic, quasiperiodic or chaotic motions, planar and three-dimensional, while only periodic planar motions have been reported for a pipe without additional mass, has motivated further studies [START_REF] Païdoussis | Non-linear dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end[END_REF][START_REF] Modarres-Sadeghi | Dynamics of cantilevered pipes conveying fluid. Part 3: Three-dimensional dynamics in the presence of an end-mass[END_REF][START_REF] Modarres-Sadeghi | Chaotic oscillations of long pipes conveying fluid in the presence of a large end-mass[END_REF]. These studies have also shown that the stability limit of the hanging flexible pipes is modified by such an end mass. Thus, with a small mass the flutter appears at a smaller critical velocity, that is a small mass is destabilizing, whereas higher masses are stabilizing [START_REF] Copeland | Chaotic flow-induced vibration of a flexible tube with end mass[END_REF][START_REF] Païdoussis | Non-linear dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end[END_REF][START_REF] Modarres-Sadeghi | Chaotic oscillations of long pipes conveying fluid in the presence of a large end-mass[END_REF]. Configurations with several additional masses [START_REF] Hill | Effects of lumped masses on the stability of fluid conveying tubes[END_REF][START_REF] Jendrzejczyk | Experiments on tubes conveying fluid[END_REF] or with a single mass at various locations along the pipe [START_REF] Sugiyama | Studies on stability of pipes conveying fluid (The effect of a lumped mass and damping)[END_REF] have also been investigated and the pipe stability has been found to depend on both the location and the value of the masses. For a comprehensive review, the reader can refer to the monography of Païdoussis [START_REF] Païdoussis | Fluid-Structure Interactions : Slender Structures and Axial Flow[END_REF]. For articulated pipes, effect of an additional mass has only be tested for horizontal pipes, rather than hanging. Thus, Sugiyama and Noda [START_REF] Sugiyama | Studies on stability of two-degree-of-freedom articulated pipes conveying fluid (Effect of an attached mass and damping)[END_REF] showed that the critical velocity for a pipe made of two segments is reduced wherever a mass is added on the second segment.

The present study aims to extend these studies to hanging articulated pipes with an arbitrary number of segments, horizontal configuration is also briefly adressed as a limit case. The effect of adding a mass on the stability of the pipe is investigated by means of a linear analysis whose formulation is derived in the next section (Sec. 2) using the Lagrange's formalism as introduced by Benjamin

Formulation of the linear analysis

The articulated pipe under investigation is schematized in Fig. 1 Following Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF], the equations of motion of the articulated pipe are derived from the n Lagrange's equations for the open system composed of the n segments and n joints, of the fluid they contain at any times and of the additional mass, that are given by d dT

∂L ∂ φi - ∂L ∂φ i = -M f U - -→ OE + U - → τ n • ∂ --→ OE ∂φ i (i = 1..n) (1)
where overdot denotes derivative with respect to time T .

In these equations L = T -V is the system Lagrangian, T = T p+f + T a is kinetic energy of the pipe and the fluid plus that of the additional mass,

V = V p+f + V a + V J
is the potential energy associated with the restoring forces due to gravity, for the pipe and the conveyed fluid, for the additional mass and due to joint stiffness. -→ τ i is the unit vector parallel to the segment i and then --→ OE + U -→ τ n is the fluid velocity at the pipe free end.

M a E (J1) (Ja) (Jn) L L a - → x - → z - → τ 1 - → τ n φ 1 φ 2 φ a φ n O U - → G
For simplicity inner and outer pipe diameters are considered as being small compared to the segment length L and the additional mass is treated as a point mass. With these assumptions and by introducing the constant angle φ 0 = 0 (then φ0 = φ0 = 0), the terms of Eq. 1 are given by

T p+f = (M p + M f ) L 3 2 n i=1   φ2 i 3 + φi i-1 j=0 φj cos (φ i -φ j ) +   i-1 j=0 φj cos (φ i -φ j )   2 +   i-1 j=0 φj sin (φ i -φ j )   2    + M f L 2 U n i=1 i-1 j=0 φj sin (φ i -φ j ) + n M f LU 2 2 , (2)
T a = M a 2 na-1 i=0   L 2 na-1 j=0 φi φj cos (φ i -φ j ) +2LL a φi φna cos (φ i -φ na ) + M a L 2 a 2 φ2 na , (3)
V p+f = -(M p + M f ) GL 2 n i=1   cos φ i 2 + i-1 j=0 cos φ j   + const., (4)
V a = -M a G na-1 i=0 L cos φ i + L a cos φ na + const. ( 5
)
The restoring forces due to the joints of stiffness K are supposed to be linearly dependent on the deformation angle in such a way that

V J = K 2 n i=1 (φ i -φ i-1 ) 2 + const. (6)
The n coupled non linear differential equations governing the motion of the articulated pipe are then derived by substituting Eqs. 2-6 into Eq. 1. Next, in order to analyze the linear stability of the straight pipe fitted with an additional mass, these equations are linearized about the equilibrium position

(φ i = 0) i=1.
.n and harmonic displacement modes φ i (T ) = ϕ i e ΣT are considered, where Σ is the complex circular frequency. The resulting linear system is nondimensionalized using the time scale Ω -1 where Ω =

3K (Mp+M f )L 3 1 2
is the frequency of the oscillator made of a single horizontal (G = 0) segment filled with the fluid at rest (U = 0). It follows the nondimensional linear homogeneous system for the complex amplitude vector ϕ = (ϕ i ) i=1..n aϕ = 0 [START_REF] Modarres-Sadeghi | Chaotic oscillations of long pipes conveying fluid in the presence of a large end-mass[END_REF] where a can be written as the sum of two n × n matrices a = a p+f + a a with a p+f the matrix for the system without additional mass and a a associated with the additional mass. Expressions of the elements a p+f (i, j) and a a (i, j) are given in Appendix, the former depend on the five nondimensional parameters n, m, g, u and σ, and the latter depend on n a , m a , l a , g and σ, where σ is the nondimensional eigenvalue σ = ΣΩ -1 . The other nondimensional parameters are defined as functions of the physical parameters as follows

n, m = 3M f M p + M f , g = 3G 2LΩ 2 , u = U LΩ , m a = 3M a (M p + M f ) L , d a = (n a -1) L + L a L = (n a -1) + l a , (8)
thus d a is the distance nondimensionalized with the segment length L from the fixed pipe end 0 to M a when the pipe is straight and l a = L a /L.

The linear stability of the static equilibrium position (φ i = 0) i=1..n of a given configuration (n, m, g, n a , d a ) is then analyzed by calculating the 2n complex eigenvalues σ of the linear problem Eq. ( 7) for a range of the nondimensional velocity u starting at 0. As u is increased, the straight pipe (φ i = 0) i=1..n is linearly destabilized as soon as one of the growth rate (σ) > 0 becomes positive. The critical velocity is given by (σ) = 0 and the pipe loses stability by divergence if the displacement mode at the threshold is static (σ) = 0, or by flutter if the displacement is oscillatory (σ) = 0. These two behaviors are illustrated in Fig. 2 showing the trajectories of the eigenvalues σ in the complex plane, as u is varied, for pipes n = 2, m = 1, g = 10 and two configurations, namely without additional (m a = 0) and with an additional mass m a = 2 at the free end d a = 2. For the configuration without additional mass, a purely real eigenvalue crosses first the imaginary axis, at a critical velocity between 5.5 eigenvalues cross first the imaginary axis, for a value of u between 4 and 5, indicating a flutter mode at the frequency given by the positive value of (σ). For some configurations, the linear displacement modes at the threshold are plotted. They are given by the real part of the solution of Eq. ( 7), associated with the first eigenvalue σ becoming positive and calculated at the critical velocity, using Matlab built-in functions. It should be noted that such linear modes are of arbitrary amplitude. For the flutter modes, the pipe displacement is shown at several instants of a cycle (see e.g. Fig. 3).

Results

Articulated pipe without additional mass

The linear stability of articulated pipes without additional mass is first analyzed by setting m a = 0 then a a is a zero matrix. In this case, the pipe stability is controlled by four nondimensional parameters, namely n, m, g and u where the mass ratio m is by definition between 0 and 3 and g is the ratio of the two restoring forces that is gravity to joint stiffness, g > 0 for hanging pipes and g = 0 corresponds to horizontal pipes. As u is increased, pipes with low g and/or low m are destabilized by flutter, otherwise they lose stability by divergence, with a boundary that depends on n (see e.g. Ref. [START_REF] Schouveiler | Flutter instability of freely hanging articulated pipes conveying fluid[END_REF]). To investigate these two behaviors, the linear analysis was mainly performed for pipes with m = 1, g = 0.5 and m = 1, g = 10, respectively referred to as "low gravity/high stiffness" and "high gravity/low stiffness" pipes. In fact, all pipes without additional mass made of n = 2 to 20 segments, as considered in the present study, and characterized by m = 1 and g = 0.5 lose stability by flutter, whereas it is by divergence if m = 1 and g = 10. Pipes made of only one segment n = 1 are always stable as discussed in Ref. [START_REF] Schouveiler | Flutter instability of freely hanging articulated pipes conveying fluid[END_REF].

The critical velocity u 0 at which articulated pipes m = 1 and g = 0.5 are destabilized, as well as the frequency ω 0 of the resulting flutter mode at u 0 , that is the value of (σ) at u 0 , are plotted as function of n in Fig. 3. The evolution of u 0 with n presents lobes which correspond to discontinuities of the flutter frequency, they are due to changes of the flutter mode. Thus, a zero-neck mode is predicted by the linear analysis for n = 2, then a one-neck mode for n = 3 and 4, a two-neck mode for n = 5, 6 and 7 and so on as illustrated in Fig. 3 for n = 2, 3, 5 and 8.

For m = 1 and g = 10, the loss of stability of the pipes leads to divergence at a critical velocity u 0 plotted in Fig. 4, u 0 weakly increases between n = 2 to 4 then is independent on n as n 4. The shape of the linear divergence modes at u 0 is shown in Fig. 4 for n = 2, 3, 5 and 8. The pipe deformation is mainly confined to the free end and closely resembles for all n. It is notable that the last segment is almost vertical, a feature already noted by Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. II. Experiments[END_REF] in his experiments who analytically shown that it would be exactly vertical (ϕ n = 0)

for pipes with joints without stiffness i.e. when gravity is the only restoring force.

Effect of an additional mass

When a mass is added on the pipe there are two more nondimensional control parameters in addition to n, m, g and u, namely the additional mass m a , which in these figures are qualitatively representative of all tested combinations such that g > 0, n up to 20 and 0 < m < 3. The case g = 0 is also briefly discussed.

"Low gravity/high stiffness" pipes

For the configurations m = 1, g = 0.5 and n = 2 to 20 segments, articulated pipes without or with an additional mass m a , whatever m a and its position d a , are found to be destabilized by flutter. Fig. 5 shows that, depending on its location along the pipe, an additional mass can have a stabilizing (u c /u 0 > 1) or destabilizing (u c /u 0 < 1) effect on the hanging articulated pipes. More precisely, the influence of the mass is null or weak when it is located on the upper part of the pipe and becomes more pronounced when the mass is on the last two segments. It is maximal on the last segment where the two extema of u c /u 0 are observed, u c /u 0 reaches its maximum around the half of this last segment and its minimum at the free end d a = n. Moreover, as shown in Figure 5(b) for n = 6, both stabilizing or destabilizing effects are greater when the mass m a is larger. For example, the critical velocity u c of a pipe n = 6, m = 1 and g = 0.5 can be increased by 1%, compared to the critical velocity of the same pipe without additional mass, when it is fitted with an additional mass m a = 1 The conclusion that a mass m a added at the free end of articulated pipes is increasingly destabilizing as m a is increased, see e.g. Fig. 5(b) for a pipe n = 6 and m a up to 8, differs from the observations for continuously flexible pipes where a small mass at the free end reduces the critical flow velocity for the flutter, whereas the effect of a higher mass is stabilizing [START_REF] Copeland | Chaotic flow-induced vibration of a flexible tube with end mass[END_REF][START_REF] Païdoussis | Non-linear dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end[END_REF][START_REF] Modarres-Sadeghi | Chaotic oscillations of long pipes conveying fluid in the presence of a large end-mass[END_REF]. For this reason, the linear stability analysis of articulated pipes was conducted with higher masses m a , up to 1000, located at the free end and u c is found to be monotonuously decreasing over the whole interval as seen for pipes n = 6, m = 1 and g = 0.5 in Fig. 7 where m a is limited to 400 for clarity. The evolution of the corresponding nondimensional flutter frequency ω c with m a is also plotted in the figure. The discontinuity that can be observed for m a between 39 and 40 corresponds to a jump from a flutter mode with 4 nodes to a mode with 5 nodes as displayed in Fig. 7.

"High gravity/low stiffness" pipes

Results for m = 1 and g = 10 are presented in Fig. 8 recalling that, for these values of the parameters, pipes without additional mass are destabilized by divergence. It appears first that adding a mass affects significantly the stability of these pipes only if it is placed on the last two segments (n -2) < d a n, and second that here also the additional mass can be stabililizing or destabililizing depending on its location. The positions for which these two effects are maximal are on the last segment, around its centre for the maximum of u c /u 0 or at its free end for its minimum, except for the n = 2 for which it is maximum at the joint between the two segments. Moreover, Fig. 8(b) shows that both stabilizing or destabilizing effects are increased as m a is increased.

The main difference with the "low gravity/high stiffness" pipes is that here adding a mass can change not only the critical velocity but also the type of the instability. Thus, when fitted with a mass, pipes m = 1 and g = 10 can be destabilized by flutter (filled symbols in Fig. 8) or divergence (open symbols), depending on the position and intensity of this mass, whereas the same pipes without additional mass are always destabilized by divergence.

To further examine this behaviour, stability analyses were conducted up to velocities such that at least one divergence mode and one flutter mode became unstable. The resulting critical velocity u c for divergence and flutter are plotted as function of the additional m a in Fig. 9 for pipes n = 6, m = 1, g = 10 fitted

with a mass at d a = 5.4 and 6.

When the mass is added at d a = 5.4 (Fig. 9(a)), the two critical velocities, for divergence and for flutter, rapidly converge towards each other as m a is increased. They intersect for m a between 2.5 and 3 meaning that the pipe is first destabilized by divergence then by flutter, the corresponding unstable modes are shown in Fig. 10(a-c) for m a = 0, 2 and 4, respectively. Note that the difference between the two critical velocities is less than 0.7% as soon as m a 2 and at least up to 100. increased it imposes a node at the free end to the flutter mode as observed in Fig. 10(f).

Horizontal pipe

Present results for hanging articulated pipes g > 0 differ from those reported by Sugiyama and Noda [START_REF] Sugiyama | Studies on stability of two-degree-of-freedom articulated pipes conveying fluid (Effect of an attached mass and damping)[END_REF] for horizontal pipes with n = 2 segments. In this configuration joint stiffness is the only restoring force then g = 0 and the pipe is destabilized by flutter, and the authors observed that wherever an additional mass is on the second segment, it is always destabilizing. Some analyses have been conducted for this horizontal configuration by setting g = 0 in the expression of the matrix elements, given in Appendix, of the linear system Eq. [START_REF] Modarres-Sadeghi | Chaotic oscillations of long pipes conveying fluid in the presence of a large end-mass[END_REF]. Results are illustrated in Fig. 11 for pipes with m = 1, n between 2 and 14, and an additional mass m a = 4. It can be seen that when the mass is positioned approximately on the first part of the pipe its impact remains limited whereas, on the second part, it is increasingly destabilizing as the mass is moved toward the free end d a = n in agreement with the results of Sugiyama and Noda. For example, for a pipe m = 1 and n = 6 fitted with a mass m a = 4, u c /u 0 evolves non-monotonously between 0.98 and 1.01 for 0 < d a 2.6 then decreases to less than 0.48 at d a = 6 that is a reduction of the critical velocity by more than 52%.

Conlusion

The present study was undertaken to investigate the effect of adding a mass on the stability of an articulated pipe conveying a fluid by means of a linear analysis. When the pipe is hanging, an additional mass can be destabilizing or stabilizing depending on its position, compared to the stability of the same pipe without additional mass, both effects increasing with the mass value. More precisely, the critical velocity of the conveyed fluid flow is found for all hanging configurations tested to be maximum when the mass is around the centre of the last segment, and minimum when it is at the free end, that is at the end of the last segment. In contrast, adding a mass on a horizontal articulated pipe has been found to be effective only to reduce the critical flow velocity at which pipe begins to flutter, this stabilizing effect occurs when m a is located roughly on the second half of the articulated and maximum at the free end.

Moreover it has been found that a sufficiently high mass added on the last two segments of "high gravity/low stiffness" articulated pipes can also change the instability type from static to oscillatory, that is from divergence to flutter. Note that similar observations have previsously been reported for another system of fluid/structure interaction. In fact, Howell et al. [START_REF] Howell | The effect of inertial inhomogeneity on the flutter of a cantileverd flexible plate[END_REF] showed that when a line mass is added parallel to the span of a flexible plate immersed in a flow, it is possible to promote as well as to delay the flow velocity at which plate flutter occurs by changing the location of the mass along the chord. Thus, the present study confirms that an "inertial inhomogeneity" [START_REF] Howell | The effect of inertial inhomogeneity on the flutter of a cantileverd flexible plate[END_REF] is an effective, and easy to implement, means to passively control the stability of a structure interacting with a fluid flow. Additional information, as the amplitude of the displacement modes or the nature of the bifurcations can be obtained by considering the system of non linear equations Eq. 1 with Eqs. 2-6. Moreover, this non linear system, possibly extended to motions not constrained in a plane, could be used to check if articulated pipes with an additionnal mass can exhibit a wide variety of dynamical states and routes to chaos as observed for a continuously flexible pipe fitted with an end mass [START_REF] Copeland | Chaotic flow-induced vibration of a flexible tube with end mass[END_REF][START_REF] Païdoussis | Non-linear dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end[END_REF][START_REF] Modarres-Sadeghi | Dynamics of cantilevered pipes conveying fluid. Part 3: Three-dimensional dynamics in the presence of an end-mass[END_REF][START_REF] Modarres-Sadeghi | Chaotic oscillations of long pipes conveying fluid in the presence of a large end-mass[END_REF].

a p+f (i, i + 1) = [9 + 6 (n -2 -i)] σ 2 2 + 2muσ -1,
and for j = 1 to i a p+f (i + 2, j) = [3 + 6 (n -2 -i)] σ 2 2 ;

if n > 3, for i = 1 to n -3 and j = i + 2 to n -1

a p+f (i, j) = [9 + 6 (n -1 -j)] σ 2 2 + 2muσ.
Remembering that n a is the number of the segment on which the additional mass mass is placed 1 n a n, elements of a a , which is a n × n symmetric matrix, are given by for all n a a a (n a , n a ) = m a σ 2 l 2 a + 2 3 gl a ;

if n a 2, for i = 1 to n a -1 a a (i, i) = m a σ 2 + 2 3 g and for i = 1 to n a -1 and j = i + 1 to n a a a (i, j) = a a (j, i) = m a σ 2 [1 + (l a -1) δ j,na ] where δ j,na is the Kronecker delta;

if n a < n, for i or j > n a a a (i, j) = 0.

[ 1 ]

 1 . Analyses were performed for articulated pipes with different characteristics and considering different mass values and positions all along the pipe. Results are presented in Section 3 before to conclude.

  . It consists of n identical rigid segments of length L and mass per unit length M p , connected by n identical flexible joints J j . The upper segment is connected in O by the joint J 1 to a pipe fixed in the frame (O, -→ x , -→ y , -→ z ). The pipe conveys a fluid, whose mass contained in one unit length of pipe is M f , at the velocity U and discharging to the atmosphere at the free end E. The pipe is fitted with an additional mass M a located on the the segment n a (1 n a n) at a distance L a (0 L a L)) from the joint J a . Without flow, the pipe freely hangs along the vertical axis (O, -→ z ) under the action of gravity G -→ z . The motions of the pipe are constrained within the plane (O, -→ x , -→ z ) in such a way that they are fully described by the n angles φ j between the segments and (O, -→ z ), its equilibrium position is defined by (φ i = 0) i=1..n . It should be pointed out that, as discussed by Benjamin [2], the conditions of stability of this equilibrium position presented in the present study are deduced from a linearized theory in such a way that they are not modified whether or not the motions of the pipe are constrained in a vertical plane.

Figure 1 :

 1 Figure 1: Schematic of the system under consideration.

Figure 2 :

 2 Figure 2: Trajectories of the eigenvalues σ in the complex plane, as u is varied from 0 to 6, for pipes n = 2, m = 1, g = 10 without additional (dashed line) and with an additional mass ma = 2 at the free end da = 2 (solid line).

Figure 3 :

 3 Figure 3: Critical velocity u 0 (•) and flutter frequency ω 0 (•) for pipes m = 1 and g = 0.5 as function of the number of segments (all pipes are destabilized by flutter) and linear flutter mode at u 0 for n = 2, 3, 5 and 8, from left to right.

Figure 4 :

 4 Figure 4: Critical velocity u 0 for pipes m = 1 and g = 10 as function of the number of segments (all pipes are destabilized by divergence) and linear divergence mode at u 0 for n = 2, 3, 5 and 8, from left to right.

Figure 5 :

 5 Figure 5: Normalized critical velocity uc/u 0 as function of the additional mass location da for m = 1 and g = 0.5 : ma = 4 and n = 2, 3, 6, 9 and 14 (a), n = 6 and ma = 0.5, 1, 2, 4 and 8 (b). All pipes are destabilized by flutter.

Fig. 6

 6 Fig.6shows the shape of the linear flutter modes for a pipe n = 6, m = 1 and g = 0.5 without or with an additional mass m a = 4 located at the positions corresponding to the extremes of the critical velocity, that is d a = 5.5 and d a = 6. At d a = 5.5, the mass has little influence on the shape of the unstable mode, in contrast, if m a is added at the free end it strongly influences by imposing a quasi-nodal point at this end, rather than an antinode without mass.

Figure 6 :

 6 Figure 6: Linear flutter mode at the critical velocity for pipe n = 6, m = 1 and g = 0.5 without additional mass at u 0 = 2.196 (a) and with a mass added at da = 5.5 and ma = 0.5, uc = 2.217 (b), ma = 4, uc = 2.431 (c) and at da = 6 and ma = 0.5, uc = 2.069 (d), ma = 4, uc = 1.386 (e).

Figure 7 :

 7 Figure 7: Critical velocity uc (•) and flutter frequency ωc (•) for pipes m = 1, g = 0.5 and n = 6 as function of the additional mass ma located at da = 6 (all pipes are destabilized by flutter) and linear flutter mode at uc for ma = 39 (left) and 40 (right).

Fig. 9 (

 9 Fig. 9(b) shows that when the mass m a is located at the free end, similar change of the type of the first mode to be destabilized occurs. In this case the two curves intersect between m a = 0.1, where the pipe loses stability by divergence Fig. 10(d), and m a = 0.2 where flutter occurs first Fig. 10(e). If m a is still

Figure 8 :

 8 Figure 8: Normalized critical velocity uc/u 0 as function of the additional mass location da for m = 1 and g = 10 : ma = 4 and n = 2, 3, 6, 9 and 14 (a), n = 6 and ma = 0.5, 1, 2, 4 and 8 (b). Open and filled symbols are for destabilizations by divergence and by flutter, respectively.

Figure 9 :

 9 Figure 9: Critical velocity uc for divergence (open symbols) and flutter (filled symbols) as function of the additional mass ma for n = 6, m = 1, g = 10 and da = 5.4 (a), da = 6 (b).

Figure 10 :

 10 Figure 10: Linearly unstable mode at the critical velocity for pipe n = 6, m = 1 and g = 10 without additional mass at u 0 = 5.938 (a) and with a mass added at da = 5.4 and ma = 2, uc = 6.954 (b), ma = 4, uc = 7.835 (c) and at da = 6 and ma = 0.1, uc = 5.982 (d), ma = 0.5, uc = 5.436 (e), ma = 4, uc = 4.189 (f).

Figure 11 :

 11 Figure 11: Normalized critical velocity uc/u 0 as function of the additional mass location da for m = 1 and g = 0, ma = 4 and n = 2, 3, 6, 9 and 14. All pipes are destabilized by flutter.

Preprint submitted to European Journal of Mechanics -B/FluidsDecember 21, 2022

σ 2 + 2muσ + mu 2 ,

Appendix

The matrix a of the linear system Eq. ( 7) is written as the sum of two n × n matrices, where n 2 is the number of segments into the pipe, a = a p+f + a a .

Elements of a p+f are given by for all n 2

and for i = 1 to n -1


François Langot
email: francois.langot@univ-lemans.fr

Rossana Merola
email: merola@ilo.org

Fabien Tripier
email: fabien.tripier@dauphine.psl.eu

Out of four major structural changes affecting the US economy -namely a rising share of skilled workers, skill-biased technological change, decreasing progressiveness of taxation and productivity slowdown -we show that the decline in productivity growth not only is the main driver of the widening wealth disparities observed in the United States of America over the past few decades, but is also the only mechanism that can explain inequalities both within and between skill groups.

List of Figures

List of Tables Table 1: Wealth inequality across skill groups in the United States: Share of unskilled households in the total population and in the four quartiles of the wealth distribution, 1989-2016 (percentage) X 1 Introduction

Over the past two decades, income inequality has increased in a large majority of member countries of the Organization for Economic Co-operation and Development. The average income gap between the top 10 per cent and the bottom 50 per cent of individuals within countries has almost doubled. If we look at wealth inequality, the wealth gap between upper-income families and middle-and lower-income families is wider than the income gap and is growing more rapidly [START_REF] Chancel | World Inequality Report 2022[END_REF]. According to a study conducted by the Pew Research Center in 2020, " [t]he richest families in the U.S. have experienced greater gains in wealth than other families in recent decades, a trend that reinforces the growing concentration of financial resources at the top" [START_REF] Horowitz | Most Americans Say There Is Too Much Economic Inequality in the US, but Fewer Than Half Call It a Top priority[END_REF]. Identifying the driving forces of inequality is a key challenge in designing effective policies to achieve inclusive and equitable growth and development.

Existing studies have pointed to several structural changes that may be leading to greater inequality. We consider four of them in this paper: declining productivity, the growing share of skilled workers in the economy, skill-based technological change, and a trend whereby taxation is becoming less progressive. With regard to the first of these processes, that is, the slowdown in productivity growth, there is clearly a discrepancy between the level of investment in the information technology sector and national productivity levels. Back in 1993, Erik Brynjolfsson referred to this situation as the "productivity paradox", which is encapsulated in the following quip: "You can see the computer age everywhere but in the productivity statistics" [START_REF] Solow | We' d Better Watch Out[END_REF]). 1 In the United States of America, the growth in labour productivity -real output per hour worked -slowed down markedly around 2004. Over the previous ten years, labour productivity in the business sector had risen at an annual average rate of more than 3 per cent. It slowed to about 2 per cent a year during 2004-10 before dropping to a paltry 0.5 per cent during 2010-16. More precisely, using data from [START_REF] Fernald | A Quarterly, Utilization-Adjusted Series on Total Factor Productivity[END_REF], we estimate that the growth rate of total factor productivity decreased significantly from 2.1 per cent before 1989 to 0.89 per cent in 2010. If the rate of return on capital, r, is much larger than the rate of growth, g, then existing capital grows faster than new capital created out of accumulated income and already rich owners of capital will become even richer. In contrast, a higher growth rate g implies higher income growth and enables those at the bottom of the distribution to accumulate new wealth, thereby reducing wealth concentration. As argued by [START_REF] Piketty | Capital in the Twenty-First Century[END_REF] and [START_REF] Piketty | Wealth and Inheritance in the Long Run[END_REF], a rate of return that exceeds economic growth (the r > g mechanism) is a key factor driving economic inequality. A second factor is related to the evolution of supply of, and demand for, skills. A rising share of skilled workers can lead to a reduction in inequalities as the relative scarcity of such workers decreases. 2 A third driver is skill-biased technological change, which pushes the skill premium up, ultimately increasing inequalities between skill groups [START_REF] Goldin | The Race Between Education and Technology[END_REF]. The fact that the skill premium is rising suggests that the skills required to handle new technologies are in scarce supply [START_REF] Acemoglu | Skills, Tasks and Technologies: Implications for Employment and Earnings[END_REF]. Moreover, it is generally agreed that skill-biased technological change led to a hollowing-out of the wage distribution in the 2000s, when many middle-wage cognitive routine occupations were automated [START_REF] Goos | Explaining Job Polarization: Routine-Biased Technological Change and Offshoring[END_REF]. The fourth driver of rising inequality has to do with the degree of progressiveness of the tax system. Over the past decade, several countries have implemented tax reforms making their tax structures much flatter (see [START_REF] Duncan | Unequal Inequalities: Do Progressive Taxes Reduce Income Inequality?[END_REF]. The shift to flat taxes, or to a tax structure with lower levels of progressiveness, is consistent with a decline in the top income tax rates and has been associated with high levels of inequality. [START_REF] Gerber | Personal Income Tax Progressivity: Trends and Implications[END_REF] find quite strong evidence of tax progressiveness having an impact on inequality.

1

The lack of diffusion of new technologies can be viewed as a major factor behind the productivity paradox [START_REF] Ernst | The AI trilemma: Saving the planet without ruining our jobs[END_REF]).

2

It is worth noting that this effect may be ambiguous. [START_REF] Aziz | Between-Group Inequality May Decline despite a Rising Skill Premium[END_REF] argue that, in societies where the share of skilled workers is below 50 per cent, a rise in educational attainment increases inequality between skill groups. Conversely, in societies where that share is above 50 per cent, an increase in educational attainment puts downward pressure on the skill premium and ultimately reduces inequality between skill groups.

To analyse the main drivers of wealth inequality, we develop a closed-economy general equilibrium model based on [START_REF] Ahn | When Inequality Matters for Macro and Macro Matters for Inequality[END_REF]. We investigate the impact of four structural changes on rising wealth inequalities: (a) the increasing gap between interest rate and productivity; (b) the rising share of educated workers; (c) skill-biased technological change; and (d) the reduction of progressiveness in taxation. In order to measure the impact of these structural changes, we consider a heterogeneous agents model à la [START_REF] Aiyagari | Uninsured Idiosyncratic Risk and Aggregate Saving[END_REF] with skilled and unskilled workers (as in [START_REF] Ahn | When Inequality Matters for Macro and Macro Matters for Inequality[END_REF], where tax distortions modify labour supply through preferences à la [START_REF] Greenwood | Investment, Capacity Utilization, and the Real Business Cycle[END_REF]. The taxation is modelled by the non-linear function proposed by [START_REF] Heathcote | Optimal Tax Progressivity: An Analytical Framework[END_REF]. Finally, in order to have the model generate wealth inequalities similar to those observed empirically, we introduce heterogeneous discount factors among workers as in [START_REF] Krusell | Income and Wealth Heterogeneity in the Macroeconomy[END_REF]. To solve and simulate the model, we employ the methodology of [START_REF] Achdou | Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach[END_REF]. The aforementioned structural changes are introduced in the model as permanent shocks that modify the steady-state wealth distribution. The shocks are calibrated using empirical data on (a) the rise in the share of skilled workers in the US labour market; (b) the technological change which favours skilled workers; (c) the decreasing progressiveness of the tax system; and (d) the slowdown in aggregate productivity growth. By comparing the initial steady state (without any shocks) and steady states in which all shocks or only one of them has occurred, we can measure the relative contribution of each shock to the rise in inequality, together with the contributions of interactions between these shocks. The modelling results show that all shocks, except for the growing share of skilled workers, induce an increase in wealth inequality between skill groups. When it comes to wealth inequality within skill groups, only the productivity slowdown can explain the surge in inequality of this kind, which is either unchanged by, or decreases in response to, shocks (a) to (c).

Our analysis contributes to inequality research in several respects. First, a very large body of literature investigating wealth inequality through general equilibrium models has emerged since the seminal contribution by [START_REF] Aiyagari | Uninsured Idiosyncratic Risk and Aggregate Saving[END_REF], with such recent important additions as Kaymak and Poschke (2016) and [START_REF] Hubmer | Sources of US Wealth Inequality: Past, Present, and Future[END_REF]. In this paper we draw on such earlier studies to identify potential drivers of wealth inequality in the United States. We extend the analytical approach taken in those studies by considering not only variations in the overall wealth distribution but also changes in inequality between and within skill groups. As noted by [START_REF] Gottschalk | Inequality, Income Growth, and Mobility: The Basic Facts[END_REF], "[i]nequality increased not only among those with different observable traits, such as gender, race, education and experience, but also within groups of workers with the same gender, race, education and experience". Building on this evidence, [START_REF] Heathcote | The Macroeconomic Implications of Rising Wage Inequality in the United States[END_REF], among others, analyse the dynamics of wage, earnings and income inequality both between and within groups. As for the role played by productivity slowdown in driving wealth inequality, 3 many studies have looked at the effect of the r > g mechanism described by Piketty (2014) (see, for example, Jones 2014; [START_REF] Krusell | Is Piketty's Second Law of Capitalism Fundamental?[END_REF][START_REF] Nardi | Piketty's Book and Macro Models of Wealth Inequality[END_REF]. To the best of our knowledge, we are the first to explore how this mechanism affects wealth inequality both between and within skill groups. Our emphasis on skills (and, therefore, also education) is justified, since the empirical evidence shows that the distribution of skills is an essential piece of the wealth-inequality puzzle [START_REF] Castañeda | Accounting for the U.S. Earnings and Wealth Inequality[END_REF][START_REF] Cooper | Household Finance over the Life-Cycle: What Does Education Contribute?[END_REF]. This ties in with the observation by [START_REF] Bartscher | Accounting for Intergenerational Income Persistence: Noncognitive Skills, Ability and Education[END_REF] that the college wealth premium in the United States has increased substantially since the 1980s and is much larger than the college income premium. There are growing concerns that, on the one hand, greater concentration of wealth may lead to bigger inheritances as the richest households pass on fortunes to their children and that, on the other hand, inheritance further increases wealth persistence [START_REF] Adermon | Intergenerational Wealth Mobility and the Role of Inheritance: Evidence from Multiple Generations[END_REF]. Existing studies suggest that inherited wealth accounts for between 15 and 31 per cent of total household wealth. 4With inherited wealth playing an increasingly greater role than self-made wealth, intergenerational social mobility may well decline. In a recent survey of around 1,000 individuals who held 3

As forcefully demonstrated by [START_REF] Piketty | Capital in the Twenty-First Century[END_REF], wealth inequality is particularly relevant because wealth is even more unequally distributed than income.

over US $ 3 million in investable assets, the Bank of America (2023) found that only 27 per cent were self made, while the rest had either some inherited wealth or an affluent upbringing or both. The negative effects of wealth concentration and inheritance on intergenerational social mobility are amplified when combined with the other advantages that wealthy parents provide their children with, such as investing more in them and sending them to better-quality schools (Blanden, Gregg, and Macmillan 2007;[START_REF] Keane | The Effect of Parental Transfers and Borrowing Constraints on Educational Attainment[END_REF][START_REF] Daruich | Explaining Intergenerational Mobility: The Role of Fertility and Family Transfers[END_REF].

Wealth concentration and inheritance could end up creating a "glass floor" below which children in wealthy families cannot fall, thereby limiting downward mobility.5

This paper is structured as follows. Section 2 presents a number of stylized facts regarding wealth inequality. Section 3 discussed the model, while section 4 explains the calibration. Section 5 discusses the results. Additional analysis considering changes in income risk over time is presented in section 6. Finally, section 7 offers some conclusions and policy implications.

X 2 Stylized Facts

The Survey of Consumer Finances (SCF) has been measuring the wealth of US households every three years since 1989. Using SFC data, earlier papers have analysed the distribution of wealth between households taking into account household characteristics such as the level of education of the family reference person (see [START_REF] Cooper | Household Finance over the Life-Cycle: What Does Education Contribute?[END_REF][START_REF] Kuhn | 2013 Update on the US Earnings, Income, and Wealth Distributional Facts: A View from Macroeconomics[END_REF][START_REF] Pfeffer | The Demography of Rising Wealth Inequality[END_REF]. 6 Recently, [START_REF] Kuhn | Income and Wealth Inequality in America, 1949-2016[END_REF] have proposed the Historical Survey of Consumer Finances (HSCF) which tracks back the distribution of wealth to 1949. [START_REF] Bartscher | Accounting for Intergenerational Income Persistence: Noncognitive Skills, Ability and Education[END_REF] report the distribution of wealth by skill groups using this survey since 1956. We complement this literature by describing extensively the dynamics of wealth inequality both between skill groups (subsection 2.1) and within skill groups (subsection 2.2).7

Inequality between skill groups

In the SFC, households are classified into four groups according to the level of education: "No high school diploma", "High school diploma", "Some college" and "College degree". Consistently with our theoretical model, we use a two-level classification (skilled and unskilled) in which the three groups with lower educational attainment ("No high school diploma", "High school diploma" and "Some college") are combined to make up the "unskilled" group, while the "skilled" group comprises those with "College degree" education. 6 [START_REF] Kuhn | 2013 Update on the US Earnings, Income, and Wealth Distributional Facts: A View from Macroeconomics[END_REF] report the distribution of earnings, income and wealth for the US economy using the 2013 SCF survey by education level (5 groups). Table 2 in [START_REF] Pfeffer | The Demography of Rising Wealth Inequality[END_REF] reports the median and Gini coefficient by education level (5 groups) for 3 waves (1989,2007,2016) of the SCF survey. Table 14 in [START_REF] Cooper | Household Finance over the Life-Cycle: What Does Education Contribute?[END_REF] shows the asset allocation for four education levels.

Table 2 documents the rise in wealth inequality across skills. The mean value of wealth increased at a rate of 2.51 per cent per year for all households raising from 353 to 689 thousands of 2016 US dollars. However, this increase in wealth has been shared very unevenly: the annual rate of wealth growth was 2.53 per cent for skilled households, compared with just 0.61 per cent for unskilled ones. The wide gap in wealth accumulation has markedly increased the wealth ratio between skilled and unskilled households from 3.35 to 5.59 over the same period. This means that unskilled households have been pushed out of the top of the wealth distribution, as can be seen in table 1. In 1989, 57 per cent of wealthy households (defined as the top quartile) were unskilled, compared with just 39 per cent in 2016. This decrease of roughly 18 percentage points is much larger than the decrease in the share of unskilled households in the total population (11 percentage points) or the decrease in the share of unskilled households in the first quartile of the wealth distribution (6 percentage points).

Inequality within skill groups

We now look at the second dimension of inequality that can be analysed using the information contained in the SCF: within-skill inequality. Table 3 compares three common indicators of inequality (the wealth Gini coefficient for skilled and unskilled, the interquartile ratio and the share of wealth possessed by the top quartile in the distribution) for all households and then by skill level, in 1989 and 2016. This table shows two striking facts: first, wealth inequality is almost identical at the global level and at the skill levels -except for the interquartile ratio which suggests a more unequal wealth distribution among unskilled households than among skilled households. Second, wealth inequality has increased both overall and in each of the skill groups. For example, the increases in the share of wealth possessed by the top quartile between 1989 and 2016 are roughly similar for all, skilled, and unskilled households: 6.24, 5.38, and 4.78 percentage points, respectively.

X Table 3

X Figure 1: Wealth and wage inequality between skills in the United States

Note: Wealth inequality is computed as the ratio of the average wealth between skilled skilled and unskilled, while wage inequality is computed as the ratio of the average wage between skilled and unskilled. Source: SCF and authors' calculations.

Figure 2 summarizes some relevant stylized facts. Between 1989 and 2016, the overall rise in wealth inequality has been driven by an increase in inequality across skill levels, as shown by the evolution of the wealth ratio in panel A, and within skill groups, as suggested by the evolution of the wealth Gini coefficients for households in the two different groups in panel B. These findings motivate the development of theoretical models with a double source of heterogeneity: agents are heterogeneous with regard to both their skill level and their labour market history, given their skill level. Both sources of heterogeneity need to be taken into account to understand the increase in wealth inequality between and within skill groups that is illustrated in figure 2. To conclude this subsection, we discuss one last stylized fact: widening wealth disparities have been highly correlated with the rise in wage inequality. Using SFC series, Figure 1 compares wealth and wage inequality between skilled and unskilled households between 1989 and 2016. Wealth inequality is twice as high as wage inequality and this ratio is remarkably stable over the period (1.99 on average, oscillating between 1.61 and 2.21 without a clear trend). The rise in wealth inequality has coincided with an increase in wage inequality (the wage ratio has almost doubled from 1.61 to 3.02). This evolution of wage inequality has been widely explained in terms of a biased technological change in favour of skilled workers8 and hence motivates the study of the role of biased technological change in shaping between-and within-skill inequality.

Is there a "wealth polarization"?

The literature on job polarization focuses on three groups of workers categorized by skill level: low-, medium-and high-skilled workers. Medium-skilled households in the United States are wealthier than unskilled ones, but experienced a fall in mean wealth between 1989 and 2016 (see table 4). Interestingly, the increase in the wealth ratio is much greater if we compare highand medium-skilled households than if we compare high-and low-skilled households (2.59 versus 1.92, respectively). When it comes to inequality within skill groups, medium-skilled households have experienced the strongest increase in the Gini coefficient of wealth distribution, as can be seen in figure 3.

X Figure 3: Wealth inequality between skills in the United States, 1989-2016

Note: Unskilled people are defined as those having at best a high-school diploma; medium-skilled people are those having at best a bachelor's degree; skilled people are those having higher than a bachelor's degree. Source: Authors' calculations based on data from the Survey of Consumer Finances.

X 3 Model

We develop a general equilibrium model with incomplete financial markets and idiosyncratic productivity shocks à la [START_REF] Achdou | Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach[END_REF]. The model is extended to include labour supply as in [START_REF] Chang | From Individual to Aggregate Labor Supply: A Quantitative Analysis Based on a Heterogeneous Agent Macroeconomy[END_REF], heterogeneous agents (skilled and unskilled workers, as in [START_REF] Ahn | When Inequality Matters for Macro and Macro Matters for Inequality[END_REF], the non-linear fiscal system à la Heathcote, Storesletten, and Violante (2017) and skill-biased technological change. Variables with a dot on top denote growth rates, while variables with a tilde on top are in level.

Households

Agents can be skilled (s) or unskilled (u) workers, q є {u, s}. Within each skill group, there exist patient (p) and impatient (i) agents characterized by their discount factor, ω є {p, i}. We denote the state vector ς = (q, ω).

The utility function of the household with skill level q is where c ~t , and h , t denote respectively consumption and hours worked and they both depend on the household's skills and patience and his/her labour market history. The parameter χ measures labour supply elasticity, γ is the risk aversion parameter and ρ is the discount factor for each type of household.

The budget constraint of this household is where  a t

, is the financial assets accumulated by households on which they receive the interest rate r t at time t. Financial markets are imperfect: there is a borrowing constraint which limits household debt to a -. At each time t, when they supply work h t , , households q receive wage  w q t , , depending on his/her skill level, which is subject to a labour income shock z q t , .

The function describing taxes T (•) is given by where τ is the tax rate on financial income, λ measures the progressiveness of the taxation scheme and g is the long-run technological progress. A fall in λ makes the fiscal system less progressive. For λ = 0 (no progressiveness), the budget constraint becomes

The process of labour income shock z Z ∈ q t q , , with z -= 1 q , is where θ q governs the persistence of the labour income shock. We will later interpret an increase in σ q as a more risky labour market for workers. The joint distributions of z q and wealth a q are denoted ( )

g a t , q q
, for q s u = { , }. . The two terms v z and v zz represent respectively the first derivative and the second derivative of the value functions with respect to the labour income shocks z q t , .

First-order conditions with respect to consumption c t and labour supply l t are respectively

The labour supply solves

Piketty effect (r-g) in the model. We consider the case without financial constraints to highlight how the (r-g) effect intervenes in the model. We omit indices for the sake of simplicity. The optimization problem yields:

where the marginal return of an asset R(a) in the stationary economy is

It declines with wealth and labour market earnings when λ > 0, with the level of taxation τ and with the long-run trend of the economy g. Therefore, a fall in long-run growth stimulates saving in the economy where γ determines the magnitude of the effect. Hence, wealth inequality will be high when individuals accumulate wealth rapidly. If we assume for simplicity that λ = 0, which implies that R(a) = (1 -τ )r -g ∀ a, a strong capitalization effect (large r -g) magnifies inequalities by facilitating wealth accumulation. The decline of the growth rate g observed since the 1990s can then rise wealth inequalities. Conversely, a decline in the interest rate, consistent with the wage premium increase, reduces wealth inequality within skill groups. It should also be noted that an increase in the tax rate on financial income τ will decrease (1 -τ )r, and thus R(a) when λ = 0 and ultimately will reduce wealth inequalities (weakening of the capitalization effect). This argument is the basis of Piketty's proposal to levy taxes on capital income to reduce inequalities (see [START_REF] Piketty | Capital in the Twenty-First Century[END_REF]. On the other hand, an objection to using capital income taxes to tackle widening inequalities is that higher taxation on capital will simply lead to capital outflows since capital is internationally mobile.

Production

The domestic production technology is based on [START_REF] Krusell | Capital-Skill Complementarity and Inequality: A Macroeconomic Analysis[END_REF] where y ~t denotes the final output, s t , and u t , the skilled and unskilled labour inputs, k ~t the physical capital input. The parameters µ ζ { , } determine the income shares of production factors. The parameter σ l is the elasticity of substitution between unskilled labour and capital or skilled labour and σ h is the elasticity of substitution between skilled labour and capital. As shown by [START_REF] Krusell | Capital-Skill Complementarity and Inequality: A Macroeconomic Analysis[END_REF], capital-skill complementarities assumes that σ σ

> l h .
We consider two sources of technological change. The first one is the technological progress e X ≡ t gt , which is common to skilled and unskilled workers (a drop in g, which can be interpreted as a global productivity slowdown). The second source is the skill-biased biased technological progress, s, defined such that a rise in s increases the demand for skilled workers and decreases the demand for unskilled workers.

Aggregation

In the financial market, the interest rate adjusts until assets held by households equal the amount of physical capital used in the production sector

In the two labour markets, skilled and unskilled q ∈ {u, s}, aggregate supply of labour equal aggregate demand for labour:

Since the size of the total population is normalized to one (in the absence of demographic growth), we impose

N N + =1 u s
and interpret an increase in N s as the outcome of higher education policy.

The level of taxation τ is adjusted to finance public spending G.

X 4 Calibration

To study the drivers of wealth inequality in the US economy, we proceed in three steps. First, the model is calibrated for the year 1989, which is the first wave of the SCF. We compute indicators of wealth inequality between and within skill groups. Second, we use external sources (e.g. observed data between 1989 and 2010) as well as the literature to calibrate the size of the shocks that may widen wealth inequalities. Third, we compute the new post-shock steady state and compare it with observed data for 2010.

Calibration of the constant parameters

X The share of skilled workers N s is calibrated so as to match the share of this population group in the SCF data in 1989. As for the parameters in the utility function, the parameter measuring agents' risk aversion ( γ ) takes the values estimated by [START_REF] Attanasio | Estimating Euler Equations[END_REF], while the elasticity of hours worked to wages (χ) is calibrated so that the model reproduces the average difference in hours worked between skilled and unskilled workers observed in the SFC for 1989.

Regarding the discount factors, (ρ q,w , for q=s, u and ω = p, i), their values and their distribution in the population of skilled and unskilled workers are calibrated so as to reproduce the values observed in the SFC for the capital ratio and other statistics summarizing the characteristics of the distribution of wealth in the United States. The parameters of the constant-elasticity-of-substitution (CES) production function take the values estimated by [START_REF] Krusell | Capital-Skill Complementarity and Inequality: A Macroeconomic Analysis[END_REF]. The growth rate of the economy and the depreciation rate are set following [START_REF] Fernald | A Quarterly, Utilization-Adjusted Series on Total Factor Productivity[END_REF] and [START_REF] Prescott | Why do Americans Work so Much More than Europeans?[END_REF]. The earning shocks are calibrated following the estimations of [START_REF] Hong | Wage Volatility and Changing Patterns of Labor Supply[END_REF]. Finally, the parameter measuring the progressiveness of the US tax system, λ, is calibrated in accordance with [START_REF] Ferrière | The Heterogeneous Effects of Government Spending: It's All About Taxes[END_REF].

Calibration of the structural changes in the US economy

Panel A of figure 4 shows the increase in the share of skilled workers in the United States between 1989 to 2016, based on SCF data. This share rises from 24 per cent to 31 per cent between 1989 and 2010 (see Table 6).

X Figure 4: Structural changes in the US economy

Panel B in figure 4 shows the historical path of the growth rate of the US economy using data from Fernald (2014). On average, the growth rate has dropped significantly from 2.1 per cent before 1989 to 0.89 per cent in 2010 (see table 6).

Based on [START_REF] Ferrière | The Heterogeneous Effects of Government Spending: It's All About Taxes[END_REF], the parameter λ has dropped significantly from 0.18 to 0.13 after 1989, reflecting a tax system which is becoming less progressive (see table 6).

X Note: SBTC = skill-biased technological change; SCF = Survey of Consumer Finances.

X 5 Identifying the sources of rising in wealth inequalities

To solve and simulate the model, we employ the methodology proposed by [START_REF] Achdou | Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach[END_REF] We calibrate the model to reproduce almost perfectly the target moments in 1989. The projections for 2010 simultaneously take into account changes in the composition of labour supply, skill-biased technical change, progressiveness of taxation and the rate of growth of the economy (see table 6). The model predicts an increase in inequality relatively close to the actual increase observed in the United States between 1989 and 2010. The Gini coefficient increases by 10 percentage points (from 0.69 to 0.79), which is very close to the 6 percentage point increase (from 0.79 to 0.85) in the empirical data. With regard to inequalities within skill groups, the share of wealth held by the richest 25 per cent remains stable in the model both for skilled and unskilled workers, while it increases very slightly in the data (from 0.82 to 0.86 for skilled workers and from 0.83 to 0.87 for unskilled workers). As for inequality between skill groups, the wealth premium -defined as the ratio of the average wealth held by skilled workers to the average wealth held by unskilled ones -increases from 3.33 to 9.12 (+5.79) in the model, while in the data the increase is more moderate, from 3.35 to 5.11 (+1.76). We calibrate our model so as to reproduce very accurately the increase in the wage premium observed in the data. The model predicts an increase in the wage premium from 2.44 in 1989 to 3.28 in 2010 (+0.84), while in the data it increases from 2.43 to 3.34 (+0.91). Therefore, we may conclude that the combination of the four structural changes is able to explain the observed evolution of inequalities in the United States from 1989 to 2010.9 Both the data and the model indicate that, since 1989, the US economy has witnessed a rise in both wealth and income inequality, although wealth seems to be more unequally distributed than wages and income, as argued by [START_REF] Piketty | Capital in the Twenty-First Century[END_REF]. Wealth inequality has increased to a greater extent between skill groups than within them.

In order to understand the impact of each of these four structural changes on the evolution of US wealth inequalities, we simulate the model by varying only one of the relevant parameters at a time. Specifically, we simulate one by one the effect of (a) an increase in the share of skilled workers; (b) technological change biased in favour of skilled workers; (c) a tax reform reducing the degree of tax progressiveness; and (d) a global productivity slowdown. These counterfactual simulations allow to isolate the impact of each factor (last four rows in Table 7).

The impact of the labour force composition

The US economy has experienced a huge increase in the share of educated workers. The "only Skill" row in table 7 shows that, according to our model, the increase in the labour supply of skilled workers relative to unskilled workers should reduce the skill wage premium from 2.44 in 1989 to 2.03 in 2010. The model predicts a fall in the skill wealth premium as well (from 3.33 to 3.21) -leaving wealth inequality within skill groups unchanged. The race between technological change and education, as argued by [START_REF] Goldin | The Race Between Education and Technology[END_REF], is the most popular explanation for growing inequality despite rising levels of education.

The change in labour force composition is not able to explain the rise in inequality observed in the data, neither between nor within skill groups. On the contrary, an increase in the share of skilled workers seems to moderate the rise in wage and wealth premium. This result is consistent with the observation in [START_REF] Aziz | Between-Group Inequality May Decline despite a Rising Skill Premium[END_REF] that in societies where the share of skilled workers is above 50 per cent, an increase in educational attainment puts downward pressures on the skill premium and ultimately reduces inequality between skill groups. The US economy crossed the 50 per cent threshold in the 1980s, which explains why an increase in the share of skilled workers has had the effect of reducing between skill inequality.

The impact of skill-biased technological change

According to a commonly held view, technological change that is biased in favour of skilled workers explains the rise in the skill wage premium (see [START_REF] Krusell | Capital-Skill Complementarity and Inequality: A Macroeconomic Analysis[END_REF]. The "only SBTC" row in table 7 shows that this shock can explain the rise in both wage and wealth premiums between skill groups. According to the empirical data, between 1989 and 2010 the skill wealth premium increased from 3.35 to 5.11 (+1.76), while the wage premium increased from 2.43 to 3.34 (+0.91). The model with an "only SBTC" shock predicts a +0.76 point increase in the skill wealth premium, from 3.33 to 4.09, and a +0.85 point increase in the skill wage premium, from 2.44 to 3.29, which means that it underestimates the increase in both wage and wealth inequality between skill groups. However, while the estimated increase in wage inequality is quite close to the empirical value, the estimated increase in wealth inequality is less than half of the value observed in the data. Moreover, with an "only SBTC" shock, our model predicts a negligible change in inequality within skill groups. Therefore, the "only SBTC" shock is unable to explain either inequality within skill groups or the observed increase in wealth inequality between skill groups.

The impact of the tax reform One of the drivers of widening inequality is the decreasing progressiveness of taxation, as demonstrated by [START_REF] Piketty | How Progressive is the US Federal Tax system? A historical and international perspective[END_REF] among others. The "only Tax" row in table 7 shows that a fiscal reform reducing the progressiveness of the tax system leads to an increase of both the wage and wealth skill premium. On the one hand, the model with the only tax shock is the able to explain both wage and wealth inequality between skill groups, although it slightly underestimates the increase in both premiums: according to the model, the wage premium should have increased by 0.24 points, whereas the empirically observed increase is 0.91 points; the model predicts a 1.45 point increase in the wealth premium, which is quite close to the increase observed in the data (1.76). In that respect, our results are close to the findings of [START_REF] Jones | The Evolution of Wealth Inequality over Half a Century: The Role of Taxes, Transfers and Technology[END_REF]. On the other hand, a less progressive tax system is unable to explain the widening of inequalities within skill groups.

The impact of growth

We finally analyse the effect of a global productivity slowdown which has been disclosed by [START_REF] Piketty | Capital in the Twenty-First Century[END_REF] and [START_REF] Piketty | Wealth and Inheritance in the Long Run[END_REF] as the r -g mechanism. The "only g" row in table 7 demonstrates the attractive nature of this mechanism: it alone is able to account for an increase in wealth inequality both between and within skill groups, as observed in the data. The model with an "only Growth" shock predicts an increase of 0.05 points in within-skill inequality for skilled workers and an increase of 0.02 points for unskilled workers. These estimates are very close to the empirically observed values, that is, a 0.04 point increase for both groups.

X 6 Extending the analysis: Changes in income risk

Beyond macroeconomic factors, structural changes can also affect microeconomic risks. Figure 5 shows that earnings risk has indeed increased since the late 1980s for both skilled and unskilled workers.

At the theoretical level, larger individual risks are not necessarily synonymous with rising wealth inequalities. Indeed, the accumulation of precautionary savings is stronger in a risky and uncertain economy and may induce a larger share of households to save. This force can then moderate the very sharp increase in wealth inequalities associated with skill-biased technical progress and economic growth. The underlying reason is that, as argued by Hubmer, Krusell, and Smith Jr (2020), higher income volatility encourages poor households to save (to protect themselves against labour market risk) and rich households to consume (since the return on assets falls).

Table 8 shows the changes in earnings risk used for our model simulations. In particular, we assume that the earnings risk increases more for unskilled workers than for skilled ones, but that its level still remains lower for the unskilled group.

Compared to results reported in table 7, table 9 shows that when we introduce income risk, the skill wealth premium decreases by 0.67 points (from 3.35 to 2.68) and the model does not account for the observed increase in the wealth ratio between skilled and unskilled workers. However, an increase in income risk has a very strong effect on wealth inequalities within skill groups. Indeed, the share of wealth held by the richest 25 per cent in both skill groups falls sharply in the model (by approximately 20 percentage points), which is counterfactual. While wage/income risk can explain higher wage or income inequality, it does not explain the contemporaneous surge in wealth inequality observed in the data. The model with the only income risk leads to a counterfactual drop in the Gini index as well.

X

X 7 Conclusions and policy implications

The stylized facts discussed in section 2 highlight the importance of understanding the drivers of rising wealth inequality between and within skill groups. Using our model to analyse the effects of various factors, we found strong evidence for the potential role of the global productivity slowdown in explaining widening wealth inequalities, both between and within skill groups.

In contrast, the other structural changes and shocks that we considered are able to explain only inequality between skill groups. Our results are consistent with the view that, while wealth and income inequality between skilled and unskilled workers is due to differences in educational attainment, within-group inequality is due to differences in productivity across firms. Workers in more successful firms are paid more than their peers with the same level of education and skills in less successful firms (see [START_REF] Barth | It's Where You Work: Increases in the Dispersion of Earnings across Establishments and Individuals in the United States[END_REF][START_REF] Furman | Slower Productivity and Higher Inequality: Are They Related?[END_REF][START_REF] Bloom | Firming Up Inequality[END_REF].

Taxation is certainly a key policy tool for sharing the gains from technological progress and reducing in-equality. Governments should not only adopt a fair tax system, but also introduce new taxes to ensure that these gains can be shared equally throughout the economy (for a discussion, see Merola 2022).

X 8 Appendix: Detailed statistics from the Survey of Consumer Finances

For our analysis we used data from the Survey of Consumer Finances (SCF), which has been measuring the wealth of US households every three years since 1989 (see [START_REF] Cooper | Household Finance over the Life-Cycle: What Does Education Contribute?[END_REF][START_REF] Kuhn | 2013 Update on the US Earnings, Income, and Wealth Distributional Facts: A View from Macroeconomics[END_REF][START_REF] Pfeffer | The Demography of Rising Wealth Inequality[END_REF].

In this appendix we report statistics from the 1989, 1995, 2001, 2004, 2007, 2010, 2013 and 2016 surveys. The following definitions are used.

• E: Earnings, defined as wages and salaries of all kinds, plus a fraction of business income.

Business income includes income from professional practices, businesses and farm sources.

The value for the fraction of business and farm income that we impute to labour earnings is the sample-wide ratio of labour income (wages plus salaries) to the sum of labour income and unambiguous capital income.

• I: Income consists of all kinds of revenue before taxes. We include government and private transfers.

• W: Wealth includes the value of financial and real assets of all kinds net of various kinds of debts. In this case, we take each component of net worth already built in the SCF; therefore, the values reported here are higher than those in [START_REF] Kuhn | 2013 Update on the US Earnings, Income, and Wealth Distributional Facts: A View from Macroeconomics[END_REF].

• Skilled: "college degree" and "postgraduate".

• Unskilled: "some college", "high school" and "dropout".

•

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 12345 Figure 1: Wealth and wage inequality between skills in the United States Figure 2: Wealth inequality across and within skill groups in the United States, 1989-2016 Figure 3: Wealth inequality between skills in the United States, 1989-2016 Figure 4: Structural changes in the US economy Figure 5: Changes in income risk in the US economy, skilled versus unskilled workers, 1968-2010

X Figure 2 :

 2 Wealth inequality across and within skill groups in the United States, 1989-2016Note: Source: Authors' calculations based on data from the Survey of Consumer Finances.

Following

  [START_REF] Achdou | Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach[END_REF], the Hamilton-Jacobi-Bellman equation in stationary variables 1 -) . The long-run technological progress g modifies the stationary discount factor (for γ ≠ 1) and the law of motion of the stationary wealth a t ,

  Notations for income distribution: B 0-1 = bottom 1 per cent; B 1-5 = bottom 1-5 per cent; B 5-10 = bottom 5-10 per cent; 1st Q = first quartile; 2nd Q = second quartile; 3rd Q = third quartile; 4th Q = fourth quartile; 5th Q = fifth quartile; T 90-95 = top 90-95 per cent; T 95-99 = top 95-99 per cent; T 99-100 = top 1 per cent.

Table 2 : Wealth inequality across skill groups in the United States: Mean value of wealth held by households, skilled versus unskilled, 1989-2016 (thousands of 2016 US dollars) Table 3: Wealth inequality within skill groups in the United States, 1989-2016 Table 4: Wealth inequality between skill groups in the United States, 1989-2016 Table 5: Calibrated parameters Table 6: Calibration of shocks Table 7: Comparison between empirical data and model results Table 8: Calibration of shocks Table 9: Comparison between empirical data and modelling results (with income risk) Table A1: Average earnings, income and wealth of US households, 1989 Table A2: Average earnings, income and wealth of US households, 1995 Table A3: Average earnings, income and wealth of US households, 2001 Table A4: Average earnings, income and wealth of US households, 2004 Table A5: Average earnings, income and wealth of US households, 2007 Table A6: Average earnings, income and wealth of US households, 2010 Table A7: Average earnings, income and wealth of US households, 2013 Table A8: Average earnings, income and wealth of US households, 2016 Table A9: Measures of inequality among US households, total and by skill group, 1989 Table A10: Measures of inequality among US households, total and by skill group, 1995 Table A11: Measures of inequality among US households, total and by skill group, 2001 Table A12: Measures of inequality among US households, total and by skill group, 2004 Table A13: Measures of inequality among US households, total and by skill group, 2007 Table A14: Measures of inequality among US households, total and by skill group, 2010 Table A15: Measures of inequality among US households, total and by skill group, 2013 Table A16: Measures of inequality among US households, total and by skill group, 2016

 2

Table 1

 1 shows that the share of unskilled households in the total population has fallen from 76.76 per cent in the first wave of the survey (1989) to 65.97 per cent in the 2016 wave, reflecting the rising level in educational attainment in the United States.

X Table

1: Wealth inequality across skill groups in the United States: Share of unskilled households in the to- tal population and in the four quartiles of the wealth distribution, 1989-2016 (percentage) All p0-25 p25-50 p50-75 p75-100

	Share of unskilled in 1989	76.76	86.83	84.68	78.77	56.75
	Share of unskilled in 2016	65.97	81.06	76.74	67.11	38.99
	Variation (in percentage points)	-10.79	-5.77	-7.93	-11.65	-17.76

Note: Source: Authors' calculations based on data from the Survey of Consumer Finances.

X Table

2:

Wealth inequality across skill groups in the United States: Mean value of wealth held by house- holds, skilled versus unskilled, 1989-2016 (thousands of 2016 US dollars)

		All	Skilled	Unskilled	Wealth Ratio
	Mean value in 1989	353.33	765.69	228.52	3.35
	Mean value in 2016	689.51	1504.17	269.29	5.59
	Annual growth rate	2.51	2.53	0.61	1.91

Note: Source: Authors' calculations based on data from the Survey of Consumer Finances.

: Wealth inequality within skill groups in the United States, 1989-2016

		All	Skilled	Unskilled
	Gini coefficient (1989)	0.79	0.75	0.77
	Gini coefficient (2016)	0.86	0.81	0.82
	Inter quartile ratio (1989)	26.16	10.16	36.26
	Inter quartile ratio (2016)	36.28	16.97	39.64
	Top quartile share (1989)	85.17	82.39	83.01
	Top quartile share (2016)	91.41	87.77	87.79

Note: Source: Authors' calculations based on data from the Survey of Consumer Finances.

Table 4 : Wealth inequality between skill groups in the United States, 1989-2016 All Skilled Medium Unskilled Ratio skilled to medium-skilled Ratio skilled to un- skilled

 4

	1989	353.33	765.69	344.84	186.50	2.22	4.11
	2016	689.51 1504.17	340.09	219.21	4.42	6.86
	Annual growth rate	2.51	2.53	-0.05	0.60	2.59	1.92

X

Note: Source: Authors' calculations based on data from Survey of Consumer Finances.

Table 5 : Calibrated parameters Category Symbol Values Sources and targets (for 1989)

 5

	Population	N s	0.24	Data (SCF 1989)
	Preferences	γ	2	Attanasio and Low (2004)
		χ	3	Target: h s /h u = 1.4
		Θ	1	Normalization
		ρ s p ,	0.0180	Target: p[25; 50] p = 0.045
		ρ s i ,	0.0535	Target: p[75; 100] p = 0.823
		ρ u p ,	0.0245	Target: p[25; 50] i = 0.032
		ρ u i ,	0.0484	Target: p[75; 100] i = 0.830
		, Share s p	1/4	Target: Capital-Output Ratio = 2.3
		Share u p ,	1/5	Target: Wealth s /Wealth u = 3.35

Table 5

 5

reports the calibrated parameter values for the initial steady state. The calibration relies on external sources and targets which are based on data available for 1989.

Table 6 : Calibration of shocks

 6

	Category	Symbol	Values (1989)	Values (2010)	Source
	SBTC	s	3.2	8.5	SCF (wage premium)
	Growth rate	g	0.021	0.0089	Fernald (2014)
	Tax	λ	0.18	0.13	Ferrière and Navarro (2018)
	Skills	N s	0.24	0.31	SCF (share of skilled households)

Comparison between empirical data and model results Wage premium Wealth premium Share of total wealth held by top 25% ( skilled workers) Share of total wealth held by top 25% (unskilled workers) Gini index

  . Results are reported in table 7. The upper part of table 7 compares empirical data with the model simulations based on the combination of the four shocks.

	Data 1989	2.43	3.35	0.82	0.83	0.79
	Model 1989	2.44	3.33	0.80	0.86	0.69
	Data 2010	3.34	5.11	0.86	0.87	0.85
	Model 2010	3.28	9.12	0.79	0.86	0.79
	Model 2010: only Skills	2.03	3.21	0.81	0.83	0.69
	Model 2010: only SBTC	3.29	4.09	0.79	0.86	0.71
	Model 2010: only Tax	2.68	4.78	0.79	0.85	0.71
	Model 2010: only Growth	2.82	6.95	0.87	0.88	0.81

X Table

7:

Table 8 : Calibration of shocks

 8

	Category	Symbol		Values (1989)	Values (2010)	Source
	Risk	{ σ σ , i 2 2 p	}	{0.0086; 0.0171}	{0.0182; 0.0285}	Hong, Seok, and You (2019)
	X

Table 9 : Comparison between empirical data and modelling results (with income risk)

 9

		Wage	Wealth	Share of total wealth	Share of total wealth	Gini index
		premium	premium	held by top 25%	held by top 25%
				(skilled workers)	(unskilled workers)
	Data 1989	2.43	3.35	0.82	0.83	0.79
	Model 1989	2.44	3.33	0.80	0.86	0.69
	Data 2010	3.34	5.11	0.86	0.87	0.85
	Model 2010	3.33	5.34	0.67	0.59	0.64
	Model 2010: only income	2.50	2.68	0.68	0.60	0.54
	risk

•

  Values are in thousands of 2016 US dollars.

	X
	ILO Working Paper 105

Table A1 : Average earnings, income and wealth of US households, 1989

 A1

			B 0-1	B 1-5	B 5-10	1st Q	2nd Q	3rd Q	4th Q	5th Q	T 90-95	T 95-99	T 99-100	All 0-100
	Skilled	E	43.10	49.77	31.78	41.24	63.98	81.82	114.76	204.99	157.30	274.75	885.88	101.07
		I	45.32	49.79	34.32	43.02	72.25	103.42	133.79	335.79	234.22	485.84	1827.06	136.94
		W	-158.25	-20.14	1.34	-0.68	133.26	404.25	903.03	5198.76	3539.53	7679.69	34124.46	1314.73
	Unskilled	E	30.23	30.55	11.28	17.49	29.82	37.55	45.72	71.30	60.22	88.87	232.02	40.39
		I	32.23	35.62	18.37	24.37	35.75	45.80	58.69	102.77	85.13	132.27	386.62	53.50
		W	-48.23	-10.23	-0.80	-4.19	25.66	124.46	317.86	1634.31	1097.43	2378.83	12081.53	419.95
	All	E	28.87	36.91	10.19	21.91	34.88	46.80	60.30	123.13	101.47	152.71	510.39	57.34
		I	30.84	40.61	17.58	27.74	41.08	56.08	77.40	181.95	132.30	231.05	997.22	76.76
		W	-84.86	-12.32	-0.63	-5.62	41.38	173.59	437.54	2712.41	1705.41	3997.51	19900.92	669.71
	X

Table A2 : Average earnings, income and wealth of US households, 1995

 A2

			B 0-1	B 1-5	B 5-10	1 st Q	2 nd Q	3 rd Q	4 th Q	5 th Q	T 90-95	T 95-99	T 99-100	All 0-100
	Skilled	E	69.74	40.79	36.34	45.45	62.62	81.95	83.94	217.28	186.12	281.56	1025.79	98.33
		I	72.93	45.09	40.18	48.91	67.11	91.42	102.60	292.22	230.78	397.73	1552.61	120.56
		W	-148.90	-16.56	5.24	4.63	102.10	253.88	579.39	3791.35	2253.95	5359.88	31136.70	947.08
	Unskilled	E	30.19	23.24	14.09	16.80	28.97	42.28	47.37	71.41	65.53	78.79	271.47	41.36
		I	33.24	27.27	19.02	22.51	34.88	49.45	57.51	99.78	87.00	111.93	464.06	52.81
		W	-78.31	-11.40	-1.06	-5.02	26.81	107.55	250.81	1270.91	805.96	1678.66	10348.00	330.15
	All	E	41.48	28.76	13.65	21.85	37.36	52.12	59.54	123.72	102.39	164.19	596.55	58.85
		I	45.11	32.57	19.89	27.20	42.87	59.24	71.52	167.74	129.91	215.33	944.82	73.62
		W	-100.06	-12.63	-0.66	-4.93	42.19	143.05	325.41	2100.46	1232.66	2751.54	18030.34	519.59
	X

Table A3: Average earnings, income and wealth of US households, 2001

	X
														ILO Working Paper 105
			B 0-1	B 1-5	B 5-10	1 st Q	2 nd Q	3 rd Q	4 th Q	5 th Q	T 90-95	T 95-99	T 99-100	All 0-100
	Skilled	E	59.58	46.43	33.37	46.27	74.76	97.43	114.12	297.35	228.34	513.18	1098.10	125.63
		I	62.34	48.54	38.11	51.01	80.51	107.81	136.51	441.81	300.87	797.51	2088.02	162.97
		W	-109.09	-21.47	5.86	12.85	160.25	408.01	933.15	5868.87	3797.41	9896.26	38571.65	1468.05
	Unskilled	E	36.31	29.01	1.44	19.80	37.19	40.99	51.03	78.45	62.27	75.14	356.80	45.48
		I	39.76	32.86	7.57	25.16	40.77	48.99	61.91	108.42	82.88	122.78	476.88	57.04
		W	-59.32	-12.11	-1.07	-4.56	23.31	97.60	265.32	1366.97	938.37	1832.35	9574.80	349.56
	All	E	50.07	33.68	11.64	24.80	43.07	53.77	72.80	169.12	135.00	261.34	776.91	72.70
		I	53.15	36.99	17.55	29.64	48.04	62.30	85.69	239.49	174.82	340.53	1444.48	93.02
		W	-77.91	-15.12	-0.44	-4.26	44.55	163.04	432.04	3013.39	1781.14	4608.47	23605.68	729.47

Table A4 : Average earnings, income and wealth of US households, 2004

 A4

			B 0-1	B 1-5	B 5-10	1st Q	2nd Q	3rd Q	4th Q	5th Q	T 90-95	T 95-99	T 99-100	All 0-100
	Skilled	E	27.16	46.16	31.43	44.79	72.61	89.69	106.65	274.15	211.79	408.06	1244.48	117.56
		I	32.97	49.03	36.08	48.96	77.95	102.38	128.67	366.34	281.90	545.45	1854.26	144.84
		W	-162.06	-29.97	-0.64	1.23	138.04	376.57	891.51	5475.11	3380.01	8891.17	38911.66	1375.86
	Unskilled	E	45.60	28.10	15.69	20.50	30.46	41.33	45.63	70.25	71.96	58.34	286.48	41.66
		I	49.85	33.82	20.50	26.05	36.04	48.90	58.36	98.81	91.83	90.67	449.15	53.67
		W	-73.78	-10.16	-0.55	-4.47	20.16	91.51	249.80	1376.79	971.83	1816.38	9957.02	347.73
	All	E	39.88	34.95	18.71	24.34	41.44	53.46	69.10	158.98	119.46	214.96	829.76	69.45
		I	44.23	39.30	23.61	29.58	46.97	62.13	83.35	213.32	150.26	293.53	1211.37	87.05
		W	-111.56	-16.33	-0.75	-6.59	40.98	157.73	426.66	3003.60	1750.45	4391.43	23950.10	724.07
	X

Table A5 : Average earnings, income and wealth of US households, 2007

 A5

			B 0-1	B 1-5	B 5-10	1st Q	2nd Q	3rd Q	4th Q	5th Q	T 90-95	T 95-99	T 99-100	All 0-100
	Skilled	E	44.21	49.97	37.40	46.12	73.28	95.54	99.26	305.01	288.20	462.22	1415.72	123.83
		I	45.27	55.06	44.24	51.61	80.52	105.72	120.68	442.35	355.60	701.57	2457.92	160.17
		W	-110.42	-28.96	1.18	3.72	149.78	387.06	857.99	5956.26	4086.98	10055.75	40267.84	1471.03
	Unskilled	E	33.19	28.13	16.21	19.92	31.57	38.67	45.03	81.88	73.09	83.11	343.55	43.40
		I	39.19	34.70	22.22	26.31	37.83	48.53	58.68	118.52	95.36	120.96	635.24	57.95
		W	-95.67	-13.05	-1.29	-6.90	19.32	95.49	264.85	1381.40	884.51	1711.94	10947.21	350.47
	All	E	38.81	36.06	17.91	24.88	38.44	53.69	69.44	172.23	117.66	282.50	875.49	71.76
		I	42.12	42.68	24.74	30.92	45.38	63.85	83.27	246.36	155.09	385.69	1509.85	93.99
		W	-107.69	-18.35	-1.29	-7.09	39.98	166.29	418.91	3106.47	1650.20	4984.13	25040.12	745.57
	X

Table A6: Average earnings, income and wealth of US households, 2010

	X
														ILO Working Paper 105
			B 0-1	B 1-5	B 5-10	1st Q	2nd Q	3rd Q	4th Q	5th Q	T 90-95	T 95-99	T 99-100	All 0-100
	Skilled	E	64.93	59.67	42.77	45.44	62.37	80.04	95.04	280.47	248.53	471.98	982.94	112.79
		I	72.13	65.31	48.21	51.68	69.15	91.69	116.94	356.70	300.16	599.34	1377.89	137.39
		W	-245.65	-70.79	-14.00	-25.51	71.81	244.93	691.27	4826.92	3287.17	7684.50	31898.93	1164.72
	Unskilled	E	51.26	37.26	25.73	22.92	25.42	38.07	42.75	65.12	57.95	88.35	242.92	38.87
		I	59.10	44.81	32.92	30.59	32.48	46.48	56.19	93.41	81.75	125.08	345.99	51.85
		W	-143.19	-32.89	-5.96	-15.16	10.64	55.22	174.33	1050.43	667.46	1474.98	8051.88	255.47
	All	E	60.42	46.40	30.27	27.87	35.53	48.19	58.25	161.79	136.45	260.55	725.57	66.43
		I	68.51	53.02	36.98	35.19	42.05	58.03	73.51	209.24	170.96	319.87	1025.27	83.74
		W	-192.15	-46.59	-7.94	-20.02	20.96	97.54	294.14	2568.44	1604.80	3982.72	20324.84	594.43

Table A7 : Average earnings, income and wealth of US households, 2013

 A7

			B 0-1	B 1-5	B 5-10	1st Q	2nd Q	3rd Q	4th Q	5th Q	T 90-95	T 95-99	T 99-100	All 0-100
	Skilled	E	44.47	44.82	38.55	37.39	59.63	76.28	96.52	273.76	227.34	399.47	1277.04	108.69
		I	70.77	50.13	44.86	44.23	67.67	90.02	121.10	396.80	296.94	571.96	2294.97	143.92
		W	-188.79	-61.00	-16.66	-21.77	64.62	236.87	624.33	4410.79	2984.98	7055.77	31757.32	1062.61
	Unskilled	E	40.15	31.24	20.80	19.38	24.79	33.19	37.87	62.54	50.57	83.38	254.23	35.56
		I	47.15	38.53	28.82	27.46	32.07	42.78	54.46	93.16	73.91	120.67	394.48	50.01
		W	-153.46	-29.95	-5.57	-14.98	9.29	47.18	151.67	954.63	578.53	1234.96	8105.64	230.00
	All	E	47.20	34.10	29.77	24.44	32.19	44.16	56.68	162.30	130.18	254.17	771.50	63.93
		I	60.41	41.18	36.36	32.20	39.60	55.19	74.97	230.38	170.34	348.23	1294.15	86.43
		W	-172.65	-41.58	-8.76	-18.63	17.95	89.28	271.00	2406.87	1337.00	3772.93	19600.45	552.93
	X

Table A8 : Average earnings, income and wealth of US households, 2016

 A8

	ILO Working Paper 105
	X
			B 0-1	B 1-5	B 5-10	1st Q	2nd Q	3rd Q	4th Q	5th Q	T 90-95	T 95-99	T 99-100	All 0-100
	Skilled	E	43.37	53.60	46.50	46.39	66.59	81.25	116.57	350.46	268.50	623.34	1658.54	132.45
		I	47.27	57.84	51.88	52.31	77.50	101.47	153.89	526.28	373.53	877.23	3012.74	182.62
		W	-207.90	-64.26	-17.16	-20.17	95.62	297.52	821.07	6165.24	3821.76	9970.48	45762.50	1476.57
	Unskilled	E	40.66	32.57	26.56	23.27	27.71	38.70	45.64	70.79	56.63	80.25	352.87	41.19
		I	48.62	39.47	33.70	31.05	35.02	48.62	63.17	110.71	87.03	136.57	514.54	57.67
		W	-97.01	-24.19	-4.79	-10.44	12.16	59.86	174.66	1135.61	653.86	1454.14	9808.87	273.68
	All	E	45.96	43.39	30.38	26.57	33.78	48.30	58.69	190.48	136.35	281.44	1118.24	71.55
		I	51.46	48.26	37.17	33.53	41.48	59.71	80.17	279.62	195.62	400.56	1799.70	98.88
		W	-149.97	-38.51	-7.20	-16.36	20.99	99.02	292.37	2990.30	1615.18	4504.90	26296.39	676.86

Table A9 : Measures of inequality among US households, total and by skill group, 1989

 A9

			Skilled			Unskilled			All
		E	I	W	E	I	W	E	I	W
	Coefficient of var-	4.51	5.04	3.84	1.17	1.47	6.28	3.85	4.60	5.06
	iation
	Varriance of logs	0.87	0.82	3.76	1.08	0.99	4.16	1.09	1.05	4.27
	Gini index	0.48	0.51	0.76	0.51	0.47	0.78	0.53	0.52	0.79
	Location of mean	68.00	74.00	79.00	61.00	64.00	77.00	65.00	69.00	80.00
	99-50 ratio	7.13	8.64	36.51	6.42	6.57	44.04	7.88	8.37	50.56
	90-50 ratio	2.53	2.48	6.62	2.56	2.34	8.31	2.63	2.61	8.50
	Mean-to-median	1.39	1.53	3.27	1.28	1.30	4.04	1.38	1.43	4.21
	ratio
	50-30 ratio	1.45	1.47	3.07	1.91	1.74	6.24	1.74	1.66	4.93
	X

Table A10 : Measures of inequality among US households, total and by skill group,

 A10

			Skilled			Unskilled			All
		E	I	W	E	I	W	E	I	W
	Coefficient of var-	3.17	4.22	5.08	1.53	2.78	5.87	2.92	4.14	5.89
	iation
	Varriance of logs	0.82	0.84	3.00	1.02	1.10	3.44	1.04	1.13	3.48
	Gini index	0.52	0.52	0.78	0.52	0.48	0.78	0.54	0.52	0.80
	Location of mean	72.00	75.00	81.00	61.00	64.00	77.00	67.00	71.00	82.00
	99-50 ratio	8.54	9.34	47.69	7.53	7.35	33.17	8.86	9.09	44.00
	90-50 ratio	2.67	2.80	7.41	2.67	2.51	6.73	2.64	2.57	7.14
	Mean-to-median	1.48	1.61	4.04	1.32	1.33	3.52	1.44	1.48	3.89
	ratio
	50-30 ratio	1.55	1.44	2.33	2.06	1.64	4.64	1.75	1.66	3.92
	X

Table A11 : Measures of inequality among US households, total and by skill group,

 A11

			Skilled			Unskilled			All
		E	I	W	E	I	W	E	I	W
	Coefficient of var-	2.15	3.18	3.94	2.01	2.06	5.03	2.34	3.40	4.94
	iation
	Varriance of logs	0.96	0.94	3.15	0.95	0.92	4.03	1.08	1.08	4.23
	Gini index	0.53	0.57	0.77	0.50	0.46	0.79	0.55	0.56	0.82
	Location of mean	75.00	79.00	81.00	64.00	64.00	78.00	69.00	73.00	81.00
	99-50 ratio	12.87	14.61	50.83	5.33	5.41	46.21	12.20	12.57	80.49
	90-50 ratio	2.62	2.84	6.85	2.58	2.46	10.02	2.86	2.79	8.91
	Mean-to-median	1.54	1.76	3.71	1.33	1.34	4.64	1.59	1.69	5.05
	ratio
	50-30 ratio	1.54	1.51	2.57	1.72	1.56	4.25	1.67	1.64	4.23
	X

Table A12 : Measures of inequality among US households, total and by skill group,

 A12

			Skilled			Unskilled			All
	99-50 ratio	9.52	10.33	52.01	5.59	5.20	40.25	9.49	10.20	80.30
	90-50 ratio	2.56	2.60	7.03	2.64	2.53	10.34	2.94	2.87	9.94
	Mean-to-median	1.46	1.55	3.94	1.27	1.31	4.34	1.54	1.59	5.41
	ratio
	50-30 ratio	1.59	1.58	2.92	1.76	1.52	5.23	1.70	1.62	4.67
	X
			Skilled			Unskilled			All
		E	I	W	E	I	W	E	I	W
	Coefficient of var-	2.45	2.78	4.03	1.56	2.05	5.33	2.53	2.99	5.11
	iation
	Varriance of logs	0.98	0.90	3.45	0.83	0.81	4.22	1.01	1.00	4.53
	Gini index	0.52	0.52	0.78	0.48	0.44	0.79	0.55	0.53	0.82
	Location of mean	70.00	73.00	81.00	62.00	63.00	78.00	68.00	70.00	81.00

Table A13 : Measures of inequality among US households, total and by skill group,

 A13

			Skilled			Unskilled			All
		E	I	W	E	I	W	E	I	W
	Coefficient of var-	2.42	3.44	4.30	3.32	4.28	5.97	2.90	4.05	5.44
	iation
	Varriance of logs	0.97	0.91	3.28	0.93	0.75	4.47	1.08	0.95	4.58
	Gini index	0.54	0.56	0.79	0.51	0.46	0.80	0.57	0.55	0.82
	Location of mean	74.00	78.00	83.00	62.00	66.00	77.00	70.00	73.00	82.00
	99-50 ratio	11.83	13.80	51.36	6.48	6.67	48.08	11.33	12.99	78.35
	90-50 ratio	2.72	2.83	7.34	2.83	2.70	9.67	2.85	2.87	8.55
	Mean-to-median	1.60	1.74	3.81	1.35	1.40	4.47	1.56	1.69	5.20
	ratio
	50-30 ratio	1.49	1.46	2.88	1.79	1.53	6.01	1.76	1.64	5.10
	X

Table A14 : Measures of inequality among US households, total and by skill group,

 A14

			Skilled			Unskilled			All
		E	I	W	E	I	W	E	I	W
	Coefficient of var-	2.59	3.24	4.55	1.84	1.96	6.72	2.74	3.41	5.88
	iation
	Varriance of logs	1.06	0.98	3.79	1.03	0.69	4.43	1.19	0.94	4.81
	Gini index	0.57	0.55	0.82	0.52	0.44	0.85	0.59	0.55	0.87
	Location of mean	73.00	75.00	81.00	61.00	64.00	79.00	71.00	73.00	83.00
	99-50 ratio	13.08	13.39	69.33	6.20	5.70	63.88	13.19	12.91	115.59
	90-50 ratio	3.23	3.19	12.27	2.84	2.60	11.91	3.06	2.96	14.20
	Mean-to-median	1.67	1.73	5.51	1.38	1.37	5.86	1.61	1.65	7.73
	ratio
	50-30 ratio	1.61	1.52	3.56	1.96	1.54	5.03	1.87	1.67	5.43
	X

Table A15: Measures of inequality among US households, total and by skill group,

	ILO Working Paper 105
			Skilled			Unskilled			All
		E	I	W	E	I	W	E	I	W
	Coefficient of var-	3.00	3.89	5.43	1.33	1.48	4.78	3.09	4.06	6.62
	iation
	Varriance of logs	1.20	1.01	3.45	1.07	0.73	4.77	1.27	0.99	4.97
	Gini index	0.58	0.57	0.82	0.53	0.45	0.87	0.60	0.56	0.87
	Location of mean	73.00	76.00	81.00	62.00	66.00	80.00	70.00	73.00	83.00
	99-50 ratio	13.63	14.24	61.29	6.80	6.36	87.15	12.96	13.62	131.06
	90-50 ratio	3.04	3.01	9.30	3.01	2.74	15.45	3.28	3.17	14.91
	Mean-to-median	1.67	1.78	4.97	1.38	1.41	7.71	1.68	1.79	8.15
	ratio
	50-30 ratio	1.69	1.59	4.24	1.98	1.55	4.41	1.93	1.66	5.48

X Table

A16: Measures of inequality among US households, total and by skill group, 2016

			Skilled			Unskilled			All
		E	I	W	E	I	W	E	I	W
	Coefficient of var-	2.49	3.46	5.30	2.08	2.29	10.01	2.75	3.87	7.30
	iation
	Varriance of logs	1.23	1.10	3.50	0.98	0.78	4.97	1.24	1.08	5.39
	Gini index	0.60	0.61	0.84	0.51	0.46	0.85	0.61	0.60	0.88
	Location of mean	75.00	79.00	83.00	63.00	66.00	82.00	72.00	77.00	85.00
	99-50 ratio	17.84	22.12	81.12	6.17	6.56	74.23	15.72	16.09	146.47
	90-50 ratio	3.14	3.36	10.35	2.74	2.47	11.27	3.39	3.22	14.09
	Mean-to-median	1.81	2.05	5.83	1.36	1.37	6.25	1.79	1.87	8.57
	ratio
	50-30 ratio	1.70	1.53	3.45	1.86	1.65	5.47	1.86	1.70	5.82

For a review of the literature on this topic, see[START_REF] Kopczuk | To Leave or Not to Leave: The Distribution of Bequest Motive[END_REF].

For further analysis of upward and downward intergenerational mobility in the United States, seeLangot et al. (2023).

Detailed statistics are reported in Appendix 7.

See, for instance,[START_REF] Krusell | Capital-Skill Complementarity and Inequality: A Macroeconomic Analysis[END_REF].

However, our model generates a too high elasticity of wealth premium.

X Figure 5: Changes in income risk in the US economy, skilled versus unskilled workers, 1968-2010

Research Department (RESEARCH)

International Labour Organization Route des Morillons 4 1211 Geneva 22 Switzerland T +41 22 799 6530 research@ilo.org www.ilo.org/research

X Advancing social justice, promoting decent work

The International Labour Organization is the United Nations agency for the world of work. We bring together governments, employers and workers to improve the working lives of all people, driving a human-centred approach to the future of work through employment creation, rights at work, social protection and social dialogue. 9 7 8 9 2 2 0 4 0 4 9 0 4 Contact details