The Dynamics of Inequality
Preliminary
Xavier Gabaix
Jean-Michel Lasry
NYU Stern
Dauphine
Pierre-Louis Lions
Benjamin Moll
Collège de France
Princeton
JRCPPF Fourth Annual Conference
February 19, 2015
1 / 25
Question
42
40
Survey of Consumer Finances
Saez and Zucman (2014)
18
38
Top 1% Wealth Share
Top 1% Income Share (excl. Capital Gains)
20
16
14
12
36
34
32
30
28
10
26
8
1950
24
1960
1970
1980
Year
1990
2000
(a) Top Income Inequality
2010
1950
1960
1970
1980
Year
1990
2000
2010
(b) Top Wealth Inequality
• In U.S. past 40 years have seen (Piketty, Saez, Zucman & coauthors)
• rapid rise in top income inequality
• rise in top wealth inequality (rapid? gradual?)
• Why?
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Question
• Main fact about top inequality (since Pareto, 1896): upper
tails of income and wealth distribution follow power laws
• Equivalently, top inequality is fractal
1
... top 0.01% are X times richer than top 0.1%,... are X times
richer than top 1%,... are X times richer than top 10%,...
2
... top 0.01% share is fraction Y of 0.1% share,... is fraction
Y of 1% share, ... is fraction Y of 10% share,...
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Evolution of “Fractal Inequality”
0.44
S(0.1)/S(1)
S(1)/S(10)
0.42
Relative Income Share
0.4
0.38
0.36
0.34
0.32
0.3
0.28
0.26
0.24
1950
•
S(p/10)
S(p)
1960
1970
1980
Year
1990
2000
2010
= fraction of top p% share going to top (p/10)%
• e.g.
S(0.1)
S(1)
= fraction of top 1% share going to top 0.1%
• Paper: same exercise for wealth
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This Paper
• Starting point: existing theories that explain top inequality
at point in time
• differ in terms of underlying economics
• but share basic mechanism for generating power laws:
random growth
• Our ultimate question: which specific economic theories can
also explain observed dynamics of top inequality?
• income: e.g. falling income taxes? superstar effects?
• wealth: e.g. falling capital taxes (rise in after-tax r − g )?
• What we do:
• study transition dynamics of cross-sectional distribution of
income/wealth in theories with random growth mechanism
• contrast with data, rule out some theories, rule in others
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Main Results
1
Transition dynamics of standard random growth models
too slow relative to those observed in the data
• analytic formula for speed of convergence
• transitions particularly slow in upper tail of distribution
2
Fast transitions require specific departures from benchmark model
• only certain economic stories generate such departures
• ⇒ eliminate the stories that cannot
3
Rise in top income inequality due to
• simple tax stories, stories about Var(permanent earnings)
• superstar effects, more complicated tax stories
4
Rise in top wealth inequality due to
• increase in r − g due to falling capital taxes
• rise in saving rates/RoRs of super wealthy
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Literature: Inequality and Random Growth
• Income distribution
• Champernowne (1953), Simon (1955), Mandelbrot (1961),
Nirei (2009), Toda (2012), Kim (2013), Jones and Kim
(2013), Aoki and Nirei (2014),...
• Wealth distribution
• Wold and Whittle (1957), Stiglitz (1969), Cowell (1998), Nirei
and Souma (2007), Benhabib, Bisin, Zhu (2012, 2014), Piketty
and Zucman (2014), Piketty and Saez (2014), Piketty (2015)
• Dynamics of income and wealth distribution
• Blinder (1973), but no Pareto tail
• Aoki and Nirei (2014)
• Power laws are everywhere ⇒ results useful there as well
• firm size distribution (e.g. Luttmer, 2007)
• city size distribution (e.g. Gabaix, 1999)
• ...
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Plan
• Theory
• a simple theory of top income inequality
• stationary distribution
• transition dynamics (this is the new stuff)
• Which economic theories can explain observed dynamics
of top inequality?
• Today’s presentation: focus on top income inequality
• Paper: analogous results for top wealth inequality
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A Random Growth Theory of Income Dynamics
• Continuous time
• Continuum of workers, heterogeneous in human capital hit
• die/retire at rate δ, replaced by young worker with hi 0
• Wage is wit = ωhit
• Human capital accumulation involves
• investment
• luck
• “Right” assumptions ⇒ wages evolve as
dwit /dt
= γit ,
wit
γit dt = γ̄dt + σdZit
• growth rate of wage wit is stochastic
• γ̄, σ depend on model parameters
√
• Zit = Brownian motion, i.e. dZit ≡ lim∆t→0 εit ∆t, εit ∼ N (0, 1)
• A number of alternative theories lead to same reduced form
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Stationary Income Distribution
• Result: The stationary income distribution has a Pareto tail
Pr(w̃ > w ) ∼ Cw −ζ
with tail inequality
1
η = = solution to quadratic equation(γ̄, σ, δ)
ζ
• Inequality η increasing in γ̄, σ, decreasing in δ
• Useful momentarily: w is Pareto ⇔ x = log w is exponential
2.5
2
slope = −ζ
Log Density, log p(x,t)
Density, f(w,t)
1.5
1
0.5
0
1
−ζ x
p(x,t)= ζ e
0
2
−2
−4
−6
−8
1.5
2
Income/Wealth, w
2.5
3
−10
0
0.5
1
1.5
2
2.5
Log Income/Wealth, x
3
3.5
4
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Transitions: The Thought Experiment
• σ ↑ leads to increase in stationary tail inequality
• But what about dynamics? Thought experiment:
• suppose economy is in Pareto steady state
• at t = 0, σ ↑. Know: in long-run → higher top inequality
2
t = 0: slope = −ξ
t = ∞: slope = −ζ
Log Density, log p(x,t)
0
−2
−4
−6
−8
−10
0
0.5
1
1.5
2
2.5
Log Income/Wealth, x
3
3.5
4
• What can we say about the speed at which this happens?
1
average speed of convergence?
2
transition in upper tail?
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Average Speed of Convergence
• Proposition: p(x, t) converges to stationary distrib. p∞ (x)
||p(x, t) − p∞ (x)|| ∼ ke −λt
with rate of convergence
1 µ2
1
+δ
2 σ 2 {µ<0}
• For given amount of top inequality η, speed λ(η, σ, δ) satisfies
λ=
∂λ
≤ 0,
∂η
∂λ
≥ 0,
∂σ
∂λ
>0
∂δ
• Observations:
• high inequality goes hand in hand with slow transitions
• half life is t1/2 = ln(2)/λ ⇒ precise quantitative predictions
• Rough idea: λ = 2nd eigenvalue of “transition matrix”
summarizing process
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Transition in Upper Tail
• So far: average speed of convergence of whole distribution
• But care in particular about speed in upper tail
• Paper: full characterization of all moments of distribution ⇒
transition can be much slower in upper tail
0
−2
Log Density, log p(x,t)
−4
−6
−8
−10
−12
−14
−16
−18
−2
t=0
t=10
t=20
t=35
Steady State Distribution
0
2
4
Log Income/Wealth, x
6
13 / 25
Dynamics of Income Inequality
• Recall process for log wages
d log wit = µdt + σdZit
+ death at rate δ
• Literature: σ has increased over last thirty years
• documented by Kopczuk, Saez and Song (2010), DeBacker et
al. (2013), Heathcote, Perri and Violante (2010) using PSID
• but Guvenen, Ozkan and Song (2014): σ stable in SSA data
• Can increase in σ explain increase in top income
inequality?
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Dynamics of Income Inequality: Model vs. Data
22
0.6
18
0.55
16
η(1)
Top 1% Labor Income Share
20
14
0.5
12
0.45
10
Data (Piketty and Saez)
Model Transition
Model Steady State
8
6
1950
2000
Year
2050
0.4
Data (Piketty and Saez)
Model Transition
Model Steady State
0.35
1950
(a) Top 1% Labor Income Share
2000
Year
2050
(b) Pareto Exponent
• Experiment σ 2 ↑ from 0.01 in 1973 to 0.025 in 2014
• Note: PL exponent η = 1 + log10
S(0.1)
S(1)
(from
S(0.1)
S(1)
= 10η−1 )
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OK, so what drives top inequality then?
Two candidates:
1
our leading example: heterogeneity in mean growth rates
2
another candidate: non-proportional random growth, i.e.
deviations from Gibrat’s law
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Heterogeneity in Mean Growth Rates
10
11
Log $’000s
12
13
14
15
(A) Mean earnings by age
25
35
45
55
age
Top 0.1%
Second 0.9%
Bottom 99%
• Guvenen, Kaplan and Song (2014): between age 25 and 35
• earnings of top 0.1% of lifetime inc. grow by ≈ 25% each year
• and only ≈ 3% per year for bottom 99%
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Heterogeneity in Mean Growth Rates
• Two regimes: H and L
dxit = µH dt + σH dZit
dxit = µL dt + σL dZit
• Assumptions
• µH > µL
• fraction θ enter labor force in H-regime
• switch from H to L at rate φ, L = absorbing state
• retire at rate δ
• Proposition: The dynamics of p̂(x, t) = E[e −ξx ] satisfy
p̂(ξ, t) − p̂∞ (ξ) = cH (ξ)e −λH (ξ)t + cL (ξ)e −λL (ξ)t
with λH (ξ) > λL (ξ), and cL (ξ), cH (ξ) = constants
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Revisiting the Rise in Income Inequality
30
0.75
0.65
0.6
20
η(1)
Top 1% Labor Income Share
0.7
25
15
0.5
0.45
10
Data (Piketty and Saez)
Model w High Growth Regime
Model Steady State
5
1950
0.55
2000
Year
(a) Top 1% Labor Income Share
2050
0.4
0.35
1950
Data (Piketty and Saez)
Model w High Growth Regime
Model Steady State
2000
Year
2050
(b) Pareto Exponent
• Experiment: in 1975 growth rate of H-types ↑ by 14%
• Empirical evidence?
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Heterogeneity in Mean Growth Rates
Some candidate economic explanations
• Different regimes = different occupations
• high growth = finance, IT,...
• Increased returns to superstars in some occupations
• larger returns to (perceived) talent
• crucial parameter: “scale of operations”, may be larger now
(ICT etc)
• Garicano and Rossi-Hansberg (2004, 2006, 2014), Gabaix and
Landier (2008)
• Could decrease in labor income taxes have played a role?
• yes, but simplest stories won’t cut it
• example of more sophisticated story: top income tax rates ↓⇒
more entry into high-growth, high-risk occupations
(“I want to be a billionaire and now it’s possible”)
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Wealth Inequality and Capital Taxes
• A simple model of top wealth inequality based on Piketty and
Zucman (2015, HID), Piketty (2015, AERPP),...
dwit = [y + (r − g − θ)wit ]dt + σwit dZit
r = (1 − τ )˜
r,
σ = (1 − τ )σ̃
• y : labor income
• Rit dt = rdt + σdZit : after-tax return on wealth
• τ : capital tax rate
• g : economy-wide growth rate
• θ: MPC out of wealth
• Stationary top inequality
η=
σ 2 /2
1
= 2
ζ
σ /2 − (r − g − θ)
• Can r − g explain observed dynamics of wealth
inequality?
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Wealth Inequality and Capital Taxes
• Compute rt − gt = r˜t (1 − τt ) − gt with details
• r˜t from Piketty and Zucman (2014)
• τt = capital tax rates from Auerbach and Hassett (2015)
• gt = smoothed growth rate from PWT
0.045
0.04
0.035
r−g
0.03
0.025
0.02
0.015
0.01
0.005
0
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
• σ = 0.3 = upper end of estimates from literature
• θ calibrated to match inequality in 1978
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Dynamics of Wealth Inequality
42
40
0.75
Data (SCF)
Data (Saez−Zucman)
Model Transition
Data (SCF)
Data (Saez−Zucman)
Model Transition
0.7
36
34
0.65
32
η
Top 1% Wealth Share
38
30
0.6
28
26
0.55
24
22
1950
1960
1970
1980
1990
Year
2000
2010
2020
2030
0.5
1950
(a) Top 1% Wealth Share
Note: PL exponent η = 1 + log10
1960
1970
1980
1990
Year
2000
2010
2020
2030
(b) Power Law Exponent
S(0.1)
S(1)
(from
S(0.1)
S(1)
= 10η−1 )
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OK, so what drives top wealth inequality then?
• Rise in rate of returns of super wealthy relative to wealthy
(top 0.01 vs. top 1%)
• better investment advice?
• better at taking advantage of “tax loopholes”?
• Rise in saving rates of super wealthy relative to wealthy
• Saez and Zucman (2014) provide some evidence
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Conclusion
• Transition dynamics of standard random growth models
too slow relative to those observed in the data
• Rise in top income inequality due to
• simple tax stories, stories about Var(permanent earnings)
• superstar effects, more complicated tax stories
• Rise in top wealth inequality due to
• increase in r − g due to falling capital taxes
• rise in saving rates/RoRs of super wealthy
25 / 25