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A Schumpeterian Model of Top Income Inequality Chad Jones and Jihee Kim SED Warsaw 2015 A Schumpeterian Model of Top Income Inequality p. 1 Top Income Inequality in the United States and France Income share of top 0.1 percent 10% 8%

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SLIDE 1

A Schumpeterian Model of Top Income Inequality

Chad Jones and Jihee Kim SED – Warsaw 2015

A Schumpeterian Model of Top Income Inequality – p. 1

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SLIDE 2

Top Income Inequality in the United States and France

1950 1960 1970 1980 1990 2000 2010 0% 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

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SLIDE 3

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) and

Rothschild 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

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SLIDE 4

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

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SLIDE 5

Top Income Inequality around the World

2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 20 22

Australia Canada Denmark France Ireland Italy Japan Mauritius New Zealand Norway 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

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SLIDE 6

The Composition of the Top 0.1 Percent Income Share

Year Top 0.1 percent income share Wages and Salaries Business income Capital income Capital gains 1950 1960 1970 1980 1990 2000 2010 0% 2% 4% 6% 8% 10% 12% 14%

A Schumpeterian Model of Top Income Inequality – p. 6

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SLIDE 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. 7

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Fractal Inequality Shares in the United States

1950 1960 1970 1980 1990 2000 2010 15 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. 8

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SLIDE 9

The Power-Law Inequality Exponent η, United States

1950 1960 1970 1980 1990 2000 2010 0.25 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. 9

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SLIDE 10

A Simple Model

A Schumpeterian Model of Top Income Inequality – p. 10

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Key Idea: Exponential growth w/ death ⇒ Pareto

Initial Time Income Creative destruction Exponential growth

A Schumpeterian Model of Top Income Inequality – p. 11

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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. 12

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What fraction of people have income greater than y?

  • Equals fraction with at least x(y) years of experience

x(y) = 1 µ log 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. 13

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SLIDE 14

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 growth

experienced 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. 14

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SLIDE 15

The Model

– Pareto distribution in partial eqm – GE with exogenous research – Full general equilibrium

A Schumpeterian Model of Top Income Inequality – p. 15

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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. 16

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Entrepreneur’s Problem – HJB Form

  • The Bellman equation for the entreprenueur:

ρV (xt, t) = max

et

log ψt+ 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 2σ2x2

t Vxx(xt, t) + Vt(xt, t)

A Schumpeterian Model of Top Income Inequality – p. 17

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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. 18

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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 existing

productivity 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. 19

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SLIDE 20
  • 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. 20

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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. 21

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SLIDE 22

Luttmer and GLLM

  • Problems with basic random growth model:
  • Luttmer (2010): Cannot produce “rockets” like Google or

Uber

  • Gabaix, Lasry, Lions, and Moll (2015): Slow transition

dynamics

  • 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. 22

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SLIDE 23

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. 23

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Growth and Creative Destruction

Final output

Y = 1

0 Y θ i di

1/θ

Production of variety i

Yi = γntxα

i Li Resource constraint

Lt + Rt + 1 = ¯ N, Lt ≡ 1

0 Litdi Flow rate of innovation

˙ nt = λ(1 − ¯ z)Rt

Creative destruction

δt = ˙ nt

A Schumpeterian Model of Top Income Inequality – p. 24

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SLIDE 25

Equilibrium with Monopolistic Competition

  • Suppose R/¯

L = ¯ s where ¯ L ≡ ¯ N − 1.

  • Define X ≡

1

0 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. 25

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Growth and Inequality in the ¯

s case

  • Creative destruction and growth

δ∗ = λR = λ(1 − ¯ z)¯ s¯ L g∗

y = ˙

n log γ = λ(1 − ¯ z)¯ s¯ L 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 ↑creative

destruction.

A Schumpeterian Model of Top Income Inequality – p. 26

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Endogenizing Research and Growth

A Schumpeterian Model of Top Income Inequality – p. 27

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Endogenizing s = R/¯

L

  • Worker:

ρV w(t) = log wt + 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. 28

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Stationary equilibrium solution

Drift of log x

˜ µ∗

H = φH(1 − τ) − β(ρ + δ∗ + ¯

δ) − 1

2σ2

H

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. 29

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SLIDE 30

Varying the x-technology parameter φ

0.3 0.4 0.5 0.6 0.7 0.25 0.50 0.75 1

POWER LAW INEQUALITY GROWTH RATE (PERCENT)

1 2 3 4

A Schumpeterian Model of Top Income Inequality – p. 30

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Why does ↑φ reduce growth?

  • ↑φ ⇒↑e∗ ⇒↑µ∗
  • Two effects
  • GE effect: technological improvement ⇒ economy more

productive 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 averse agents undertake less research.

  • Positive level effect raises both profits and wages. Riskier

research ⇒ lower research and lower long-run growth.

A Schumpeterian Model of Top Income Inequality – p. 31

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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 in

medium run

A Schumpeterian Model of Top Income Inequality – p. 32

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SLIDE 33

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. 33

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Possible explanations: Inequality in France

  • Efficiency-reducing explanations
  • Delayed adoption of good technologies (WWW)
  • Increased misallocation (killing off entrepreneurs more

quickly)

  • Efficiency-enhancing explanations
  • Increased subsidies to research (more creative

destruction)

  • Reduction in blocking of innovations (more creative

destruction)

A Schumpeterian Model of Top Income Inequality – p. 34

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SLIDE 35

Micro Evidence

A Schumpeterian Model of Top Income Inequality – p. 35

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Overview

  • Geometric random walk with drift = canonical DGP in the

empirical 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. 36

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SLIDE 37

Pareto Tails of the Growth Rate Distribution

From Guvenen et al (2015)

⇒ ˜ µH ⇒ ¯ δ + δ

A Schumpeterian Model of Top Income Inequality – p. 37

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SLIDE 38

Growth Rates of Top 5% Incomes, 1988–1989

−4 −3 −2 −1 1 2 20 40 60 80 100 120 140 160 180 Change in log income Number of observations

A Schumpeterian Model of Top Income Inequality – p. 38

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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. 39

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SLIDE 40

Three numerical examples

A Schumpeterian Model of Top Income Inequality – p. 40

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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 structural

parameters, e.g. φ and τ

  • Sequence of steady states, not transition dynamics

A Schumpeterian Model of Top Income Inequality – p. 41

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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 inequality

in 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. 42

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SLIDE 43

Numerical Example: Matching U.S. Inequality

1980 1985 1990 1995 2000 2005 0.2 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 INEQUALITY GROWTH RATE (PERCENT)

1.0 1.5 2.00 2.5 3.0

A Schumpeterian Model of Top Income Inequality – p. 43

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SLIDE 44

Numerical Example: U.S. and France

1980 1985 1990 1995 2000 2005 0.2 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 INEQUALITY GROWTH RATE (PERCENT)

1.0 1.5 2.00 2.5 3.0

A Schumpeterian Model of Top Income Inequality – p. 44

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SLIDE 45

Numerical Example: Taxes and Inequality

1980 1985 1990 1995 2000 2005 0.2 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 INEQUALITY GROWTH RATE (PERCENT)

1.0 1.5 2.00 2.5 3.0

A Schumpeterian Model of Top Income Inequality – p. 45

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SLIDE 46

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: raises

creative 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. 46

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SLIDE 47

Extra Slides

A Schumpeterian Model of Top Income Inequality – p. 47

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SLIDE 48

Overview

  • Atkinson / Piketty / Saez stylized facts on top income

inequality

  • Rising sharply in US since 1980
  • More stable in France and Japan
  • Why?

A Schumpeterian Model of Top Income Inequality – p. 48

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SLIDE 49

The Pareto Nature of Labor Income

$0 $500k $1.0m $1.5m $2.0m $2.5m $3.0m 1 2 3 4 5 6 7 8 9 Wage income cutoff, z Income ratio: Mean( y | y>z ) / z 2005 1980

Equals

1 1−η if Pareto...

A Schumpeterian Model of Top Income Inequality – p. 49

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SLIDE 50

Skill-Biased Technical Change?

  • Let xi = skill and ¯

w = wage per unit skill yi = ¯ wxα

i

  • 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. 50

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SLIDE 51

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)]

  • utflow 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. 51

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SLIDE 52

How the model works

  • ↑φ raises top inequality, but leaves the growth rate of the

economy 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. 52