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A Schumpeterian Model of
Top Income Inequality
Chad Jones and Jihee Kim
A Schumpeterian Model of Top Income Inequality – p. 1
Top Income Inequality in the United States and France
1950 1960 1970 1980 1990 2000 20100%
2%
4%
6%
8%
10%
Year
Income share of top 0.1 percent
United States
France
Source: World Top Incomes Database (Alvaredo, Atkinson, Piketty, Saez)
A Schumpeterian Model of Top Income Inequality – p. 2
Related literature
• Empirics: Piketty and Saez (2003), Aghion et al (2015),Guvenen-Kaplan-Song (2015) and many more
• Rent Seeking: Piketty, Saez, and Stantcheva (2011) andRothschild and Scheuer (2011)
• Finance: Philippon-Reshef (2009), Bell-Van Reenen (2010)
• Not just finance: Bakija-Cole-Heim (2010), Kaplan-Rauh
• Pareto-generating mechanisms: Gabaix (1999, 2009),Luttmer (2007, 2010), Reed (2001). GLLM (2015).
• Use Pareto to get growth: Kortum (1997), Lucas and Moll(2013), Perla and Tonetti (2013).
• Pareto wealth distribution: Benhabib-Bisin-Zhu (2011), Nirei(2009), Moll (2012), Piketty-Saez (2012), Piketty-Zucman(2014), Aoki-Nirei (2015)
A Schumpeterian Model of Top Income Inequality – p. 3
Outline
• Facts from World Top Incomes Database
• Simple model
• Full model
• Empirical work using IRS public use panel tax returns
• Numerical examples
A Schumpeterian Model of Top Income Inequality – p. 4
Top Income Inequality around the World
2 4 6 8 10 12 14 16 182
4
6
8
10
12
14
16
18
20
22
Australia
Canada
Denmark
France
Ireland
Italy Japan
Mauritius
New ZealandNorway
Singapore
Spain
Sweden
Switzerland
United States
Top 1% share, 1980−82
Top 1% share, 2006−08
45−degree line
A Schumpeterian Model of Top Income Inequality – p. 5
The Composition of the Top 0.1 Percent Income Share
Year
Top 0.1 percent income share
Wages and Salaries
Businessincome
Capital income
Capital gains
1950 1960 1970 1980 1990 2000 20100%
2%
4%
6%
8%
10%
12%
14%
A Schumpeterian Model of Top Income Inequality – p. 6
The Pareto Nature of Labor Income
$0 $500k $1.0m $1.5m $2.0m $2.5m $3.0m1
2
3
4
5
6
7
8
9
Wage income cutoff, z
Income ratio: Mean( y | y>z ) / z
2005
1980
Equals 1
1−ηif Pareto...
A Schumpeterian Model of Top Income Inequality – p. 7
Pareto Distributions
Pr [Y > y] =
(y
y0
)−ξ
• Let S(p) = share of income going to the top p percentiles,
and η ≡ 1/ξ be a measure of Pareto inequality:
S(p) =
(100
p
)η−1
If η = 1/2, then share to Top 1% is 100−1/2 ≈ .10
If η = 3/4, then share to Top 1% is 100−1/4 ≈ .32
• Fractal: Let S(a) = share of 10a’s income going to top a:
S(a) = 10η−1
A Schumpeterian Model of Top Income Inequality – p. 8
Fractal Inequality Shares in the United States
1950 1960 1970 1980 1990 2000 201015
20
25
30
35
40
45
S(1)
S(.1)
S(.01)
Year
Fractal shares (percent)
From 20% in 1970 to 35% in 2010
A Schumpeterian Model of Top Income Inequality – p. 9
The Power-Law Inequality Exponent η, United States
1950 1960 1970 1980 1990 2000 20100.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
η(1)
η(.1)
η(.01)
Year
1 + log10
(top share)
η rises from .33 in 1970 to .55 in 2010
A Schumpeterian Model of Top Income Inequality – p. 10
Skill-Biased Technical Change?
• Let xi = skill and w = wage per unit skill
yi = wxαi
• If Pr [xi > x] = x−1/ηx , then
Pr [yi > y] =( y
w
)−1/ηy
where ηy = αηx
• That is yi is Pareto with inequality parameter ηy
SBTC (↑ w) shifts distribution right but ηy unchanged.
↑α would raise Pareto inequality...
This paper: why is x ∼ Pareto, and why ↑α
A Schumpeterian Model of Top Income Inequality – p. 11
A Simple Model
Cantelli (1921), Steindl (1965), Gabaix (2009)
A Schumpeterian Model of Top Income Inequality – p. 12
Key Idea: Exponential growth w/ death ⇒ Pareto
TIME
INCOME
Initial
Creative
destructionExponential
growth
A Schumpeterian Model of Top Income Inequality – p. 13
Simple Model for Intuition
• Exponential growth often leads to a Pareto distribution.
• Entrepreneurs
New entrepreneur (“top earner”) earns y0
Income after x years of experience:
y(x) = y0eµx
• Poisson “replacement” process at rate δ
Stationary distribution of experience is exponential
Pr [Experience > x] = e−δx
A Schumpeterian Model of Top Income Inequality – p. 14
What fraction of people have income greater than y?
• Equals fraction with at least x(y) years of experience
x(y) =1
µlog
(y
y0
)
• Therefore
Pr [Income > y] = Pr [Experience > x(y)]
= e−δx(y)
=
(y
y0
)−
δ
µ
• So power law inequality is given by
ηy =µ
δ
A Schumpeterian Model of Top Income Inequality – p. 15
Intuition
• Why does the Pareto result emerge?
Log of income ∝ experience (Exponential growth)
Experience ∼ exponential (Poisson process)
Therefore log income is exponential
⇒ Income ∼ Pareto!
• A Pareto distribution emerges from exponential growthexperienced for an exponentially distributed amount of time.
Full model: endogenize µ and δ and how they change
A Schumpeterian Model of Top Income Inequality – p. 16
Why is experience exponentially distributed?
• Let F (x, t) denote the distribution of experience at time t
• How does it evolve over discrete interval ∆t?
F (x, t+∆t)− F (x, t) = δ∆t(1− F (x, t))︸ ︷︷ ︸
inflow from above x
− [F (x, t)− F (x−∆x, t)]︸ ︷︷ ︸
outflow as top folks age
• Dividing both sides by ∆t = ∆x and taking the limit
∂F (x, t)
∂t= δ(1− F (x, t))−
∂F (x, t)
∂x
• Stationary: F (x) such that∂F (x,t)∂t = 0. Integrating gives the
exponential solution.
A Schumpeterian Model of Top Income Inequality – p. 17
The Model
– Pareto distribution in partial eqm– GE with exogenous research– Full general equilibrium
A Schumpeterian Model of Top Income Inequality – p. 18
Entrepreneur’s Problem
Choose et to maximize expected discounted utility:
U(c, ℓ) = log c+ β log ℓ
ct = ψtxt
et + ℓt + τ = 1
dxt = µ(et)xtdt+ σxtdBt
µ(e) = φe
x = idiosyncratic productivity of a variety
ψt = determined in GE (grows)
δ = endogenous creative destruction
δ = exogenous destruction
A Schumpeterian Model of Top Income Inequality – p. 19
Entrepreneur’s Problem – HJB Form
• The Bellman equation for the entreprenueur:
ρV (xt, t) = maxet
logψt+ log xt + β log(Ω− et) +E[dV (xt, t)]
dt
+(δ + δ)(V w(t)− V (xt, t))
where Ω ≡ 1− τ
• Note: the “capital gain” term is
E[dV (xt, t)]
dt= µ(et)xtVx(xt, t) +
1
2σ2x2tVxx(xt, t) + Vt(xt, t)
A Schumpeterian Model of Top Income Inequality – p. 20
Solution for Entrepreneur’s Problem
• Equilibrium effort is constant:
e∗ = 1− τ −1
φ· β(ρ+ δ + δ)
• Comparative statics:
↑τ ⇒↓e∗: higher “taxes”
↑φ⇒↑e∗: better technology for converting effort into x
↑δ or δ ⇒↓e∗: more destruction
A Schumpeterian Model of Top Income Inequality – p. 21
Stationary Distribution of Entrepreneur’s Income
• Unit measure of entrepreneurs / varieties
• Displaced in two ways
Exogenous misallocation (δ): new entrepreneur → x0.
Endogenous creative destruction (δ): inherit existingproductivity x.
• Distribution f(x, t) satisfies Kolmogorov forward equation:
∂f(x, t)
∂t= −δf(x, t)−
∂
∂x[µ(e∗)xf(x, t)] +
1
2·∂2
∂x2[σ2x2f(x, t)
]
• Stationary distribution limt→∞ f(x, t) = f(x) solves∂f(x,t)∂t = 0
A Schumpeterian Model of Top Income Inequality – p. 22
• Guess that f(·) takes the Pareto form f(x) = Cx−ξ−1 ⇒
ξ∗ = −µ∗
σ2+
√(µ∗
σ2
)2
+2 δ
σ2
µ∗ ≡ µ(e∗)−1
2σ2 = φ(1− τ)− β(ρ+ δ∗ + δ)−
1
2σ2
• Power-law inequality is therefore given by
η∗ = 1/ξ∗
A Schumpeterian Model of Top Income Inequality – p. 23
Comparative Statics (given δ∗)
η∗ = 1/ξ∗, ξ∗ = −µ∗
σ2+
√(µ∗
σ2
)2
+2 δ
σ2
µ∗ = φ(1− τ)− β(ρ+ δ∗ + δ)−1
2σ2
• Power-law inequality η∗ increases if
↑φ: better technology for converting effort into x
↓ δ or δ: less destruction
↓ τ : Lower “taxes”
↓ β: Lower utility weight on leisure
A Schumpeterian Model of Top Income Inequality – p. 24
Luttmer and GLLM
• Problems with basic random growth model:
Luttmer (2011): Cannot produce “rockets” like Google orUber
Gabaix, Lasry, Lions, and Moll (2015): Slow transitiondynamics
• Solution from Luttmer/GLLM:
Introduce heterogeneous mean growth rates: e.g. “high”versus “low”
Here: φH > φL with Poisson rate p of transition (H → L)
A Schumpeterian Model of Top Income Inequality – p. 25
Pareto Inequality with Heterogeneous Growth Rates
η∗ = 1/ξH, ξH = −µ∗
H
σ2+
√(µ∗
H
σ2
)2
+2 (δ + p)
σ2
µ∗H= φH(1− τ)− β(ρ+ δ∗ + δ)−
1
2σ2
• This adopts Gabaix, Lasry, Lions, and Moll (2015)
• Why it helps quantitatively:
φH: Fast growth allows for Google / Uber
p: Rate at which high growth types transit to low growth
types raises the speed of convergence = δ + p.
A Schumpeterian Model of Top Income Inequality – p. 26
Growth and Creative Destruction
Final output Y =(∫ 1
0 Yθi di
)1/θ
Production of variety i Yi = γntxαi Li
Resource constraint Lt +Rt + 1 = N , Lt ≡∫ 10 Litdi
Flow rate of innovation nt = λ(1− z)Rt
Creative destruction δt = nt
A Schumpeterian Model of Top Income Inequality – p. 27
Equilibrium with Monopolistic Competition
• Suppose R/L = s where L ≡ N − 1.
• Define X ≡∫ 10 xidi =
x0
1−η . Markup is 1/θ.
Aggregate PF Yt = γntXαL
Wage for L wt = θγntXα
Profits for variety i πit = (1− θ)γntXαL(xi
X
)∝ wt
(xi
X
)
Definition of ψt ψt = (1− θ)γntXα−1L
Note that ↑η has a level effect on output and wages.
A Schumpeterian Model of Top Income Inequality – p. 28
Growth and Inequality in the s case
• Creative destruction and growth
δ∗ = λR = λ(1− z)sL
g∗y = n log γ = λ(1− z)sL log γ
• Does rising top inequality always reflect positive changes?
No! ↑ s (more research) or ↓ z (less innovation blocking)
Raise growth and reduce inequality via ↑creativedestruction.
A Schumpeterian Model of Top Income Inequality – p. 29
Endogenizing Research
and Growth
A Schumpeterian Model of Top Income Inequality – p. 30
Endogenizing s = R/L
• Worker:
ρV w(t) = logwt +dV W (t)
dt
• Researcher:
ρV R(t) = log(mwt) +dV R(t)
dt+ λ
(E[V (x, t)]− V R(t)
)
+ δR(V (x0, t)− V R(t)
)
• Equilibrium: V w(t) = V R(t)
A Schumpeterian Model of Top Income Inequality – p. 31
Stationary equilibrium solution
Drift of log x µ∗H= φH(1− τ)− β(ρ+ δ∗ + δ)− 1
2σ2H
Pareto inequality η∗ = 1/ξ∗, ξ∗ = − µ∗
H
σ2
H
+
√(µ∗
H
σ2
H
)2+ 2 (δ+p)
σ2
H
Creative destruction δ∗ = λ(1− z)s∗L
Growth g∗ = δ∗ log γ
Research allocation V w(s∗) = V R(s∗)
A Schumpeterian Model of Top Income Inequality – p. 32
Varying the x-technology parameter φ
0.3 0.4 0.5 0.6 0.70
0.25
0.50
0.75
1
POWER LAW INEQUALITYGROWTH RATE (PERCENT)
0
1
2
3
4
A Schumpeterian Model of Top Income Inequality – p. 33
Why does ↑φ reduce growth?
• ↑φ⇒↑e∗ ⇒↑µ∗
• Two effects
GE effect: technological improvement ⇒economy moreproductive so higher profits, but also higher wages
Allocative effect: raises Pareto inequality (η), so xi
X is
more dispersed ⇒E log πi/w is lower. Risk averseagents undertake less research.
• Positive level effect raises both profits and wages. Riskierresearch ⇒ lower research and lower long-run growth.
A Schumpeterian Model of Top Income Inequality – p. 34
How the model works
• ↑φ raises top inequality, but leaves the growth rate of theeconomy unchanged.
Surprising: a “linear differential equation” for x.
• Key: the distribution of x is stationary!
• Higher φ has a positive level effect through higher inequality,raising everyone’s wage.
But growth comes via research, not through x...
Lucas at “micro” level, Romer/AH at “macro” level
A Schumpeterian Model of Top Income Inequality – p. 35
Growth and Inequality
• Growth and inequality tend to move in opposite directions!
• Two reasons
1. Faster growth ⇒more creative destruction
Less time for inequality to grow
Entrepreneurs may work less hard to grow market
2. With greater inequality, research is riskier!
Riskier research ⇒ less research ⇒ lower growth
• Transition dynamics ⇒ambiguous effects on growth inmedium run
A Schumpeterian Model of Top Income Inequality – p. 36
Possible explanations: Rising U.S. Inequality
• Technology (e.g. WWW)
Entrepreneur’s effort is more productive ⇒↑η
Worldwide phenomenon, not just U.S.
Ambiguous effects on U.S. growth (research is riskier!)
• Lower taxes on top incomes
Increase effort by entrepreneur’s ⇒↑η
A Schumpeterian Model of Top Income Inequality – p. 37
Possible explanations: Inequality in France
• Efficiency-reducing explanations
Delayed adoption of good technologies (WWW)
Increased misallocation (killing off entrepreneurs morequickly)
• Efficiency-enhancing explanations
Increased subsidies to research (more creativedestruction)
Reduction in blocking of innovations (more creativedestruction)
A Schumpeterian Model of Top Income Inequality – p. 38
Micro Evidence
A Schumpeterian Model of Top Income Inequality – p. 39
Overview
• Geometric random walk with drift = canonical DGP in theempirical literature on income dynamics.
– Survey by Meghir and Pistaferri (2011)
• The distribution of growth rates for the Top 10% earners
Guvenen, Karahan, Ozkan, Song (2015) for 1995-96
IRS public use panel for 1979–1990 (small sample)
A Schumpeterian Model of Top Income Inequality – p. 40
Growth Rates of Top 10% Incomes, 1995–1996
-5 -4 -3 -2 -1 0 1 2 3 40
1
2
3
4
5
6
1 i
n 1
00:
rise
by a
fac
tor
of
3.0
1 i
n 1
,000:
rise
by a
fac
tor
of
6.8
1 i
n 1
0,0
00:
rise
by a
fac
tor
of
24.6
ANNUAL LOG CHANGE, 1995-96
DENSITY
︷ ︸︸ ︷
⇒ µH
⇒ δ + δ︷ ︸︸ ︷
From Guvenen et al (2015)
A Schumpeterian Model of Top Income Inequality – p. 41
Growth Rates of Top 5% Incomes, 1988–1989
−4 −3 −2 −1 0 1 20
20
40
60
80
100
120
140
160
180
Change in log income
Number of observations
A Schumpeterian Model of Top Income Inequality – p. 42
Results
IRS IRS Guvenen et al.Parameter 1979–81 1988–90 1995–96
δ + δ 0.07 ...
σH 0.122 ...
p 0.767 ...
µH 0.244 0.303 0.435
Model: η∗ 0.330 0.398 0.556
Data: η 0.33 0.48 0.55
A Schumpeterian Model of Top Income Inequality – p. 43
Three numerical examples
A Schumpeterian Model of Top Income Inequality – p. 44
Three numerical examples
• The examples
1. Match U.S. inequality 1980–2007 (φ)
2. Match inequality in France (z, p)
3. Match U.S. and French data using taxes (τ)
• Why these are just examples
Identification problem: observe µ but not structuralparameters, e.g. φ and τ
Sequence of steady states, not transition dynamics
A Schumpeterian Model of Top Income Inequality – p. 45
Parameters
• Parameters consistent with IRS panel:
φ ≈ 0.5 ⇒ µH ≈ .3
σH = σL = .122
p = 0.767
q = .504 – 2.5% of top earners are high growth
• Other parameter values
Match U.S. growth of 2% per year and Pareto inequalityin 1980
δ = 0.04 and γ = 1.4 ⇒ δ + δ ≈ 0.10
ρ = 0.03, L = 15, τ = 0, θ = 2/3, β = 1, λ = 0.027, m =0.5, z = 0.20
A Schumpeterian Model of Top Income Inequality – p. 46
Numerical Example: Matching U.S. Inequality
1980 1985 1990 1995 2000 20050.2
0.3
0.4
0.5
0.6
φH in US rises from 0.385 to 0.568
US Growth
(right scale) US, η
(left scale)
POWER LAW INEQUALITYGROWTH RATE (PERCENT)
1.0
1.5
2.00
2.5
3.0
A Schumpeterian Model of Top Income Inequality – p. 47
Numerical Example: U.S. and France
1980 1985 1990 1995 2000 20050.2
0.3
0.4
0.5
0.6
p in France rises from 0.89 to 1.09
z in France falls from 0.350 to 0.250
France, η
US, η
POWER LAW INEQUALITYGROWTH RATE (PERCENT)
1.0
1.5
2.00
2.5
3.0
A Schumpeterian Model of Top Income Inequality – p. 48
Numerical Example: Taxes and Inequality
1980 1985 1990 1995 2000 20050.2
0.3
0.4
0.5
0.6
τ in France falls from 0.395 to 0.250
τ in the U.S. falls from 0.350 to 0.038
France, η
US, η
POWER LAW INEQUALITYGROWTH RATE (PERCENT)
1.0
1.5
2.00
2.5
3.0
A Schumpeterian Model of Top Income Inequality – p. 49
Conclusions: Understanding top income inequality
• Information technology / WWW:
Entrepreneurial effort is more productive: ↑φ⇒↑η
Worldwide phenomenon (?)
• Why else might inequality rise by less in France?
Less innovation blocking / more research: raisescreative destruction
Regulations limiting rapid growth: ↑ p and ↓φ
Theory suggests rich connections between:
models of top inequality ↔ micro data on income dynamics
A Schumpeterian Model of Top Income Inequality – p. 50