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 Income share of top 0.1 percent 10% 8% United States 6% 4% France 2% 0% 1950 1960 1970 1980 1990 2000 2010 Year 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) 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
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 Top 1% share, 2006−08 22 20 18 United States 16 Singapore 14 Canada 45−degree line 12 Ireland Switzerland 10 Italy Japan Australia France Spain Norway 8 New Zealand Sweden Mauritius 6 Denmark 4 2 2 4 6 8 10 12 14 16 18 Top 1% share, 1980−82 A Schumpeterian Model of Top Income Inequality – p. 5
The Composition of the Top 0.1 Percent Income Share Top 0.1 percent income share 14% 12% 10% Capital gains 8% 6% Business 4% income 2% Capital income Wages and Salaries 0% 1950 1960 1970 1980 1990 2000 2010 Year A Schumpeterian Model of Top Income Inequality – p. 6
The Pareto Nature of Labor Income Income ratio: Mean( y | y>z ) / z 9 8 7 1 Equals 1 − η if Pareto... 6 5 4 3 2005 2 1980 1 $0 $500k $1.0m $1.5m $2.0m $2.5m $3.0m Wage income cutoff, z A Schumpeterian Model of Top Income Inequality – p. 7
Pareto Distributions � y � − ξ Pr [ Y > y ] = y 0 • Let ˜ S ( p ) = share of income going to the top p percentiles, and η ≡ 1 /ξ be a measure of Pareto inequality: � 100 � η − 1 ˜ S ( p ) = p ◦ 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 10 a ’s income going to top a : S ( a ) = 10 η − 1 A Schumpeterian Model of Top Income Inequality – p. 8
Fractal Inequality Shares in the United States Fractal shares (percent) 45 40 S(.01) From 20% in 1970 to 35% in 2010 35 30 S(1) 25 S(.1) 20 15 1950 1960 1970 1980 1990 2000 2010 Year A Schumpeterian Model of Top Income Inequality – p. 9
The Power-Law Inequality Exponent η , United States 1 + log 10 (top share) 0.65 0.6 η (.01) 0.55 η rises from .33 in 1970 to .55 in 2010 0.5 0.45 η (1) 0.4 η (.1) 0.35 0.3 0.25 1950 1960 1970 1980 1990 2000 2010 Year A Schumpeterian Model of Top Income Inequality – p. 10
Skill-Biased Technical Change? • Let x i = skill and ¯ w = wage per unit skill wx α y i = ¯ i • If Pr [ x i > x ] = x − 1 /η x , then � y � − 1 /η y Pr [ y i > y ] = where η y = αη x w ¯ • That is y i is Pareto with inequality parameter η y ◦ SBTC ( ↑ ¯ w ) shifts distribution right but η y 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 INCOME Creative Exponential destruction growth Initial TIME 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 y 0 ◦ Income after x years of experience: y ( x ) = y 0 e µx • 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 � y � x ( y ) = 1 µ log y 0 • Therefore Pr [ Income > y ] = Pr [ Experience > x ( y )] = e − δx ( y ) � y � − δ µ = y 0 • So power law inequality is given by η y = 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 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. 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 )) − [ F ( x, t ) − F ( x − ∆ x, t )] � �� � � �� � inflow from above x outflow as top folks age • Dividing both sides by ∆ t = ∆ x and taking the limit ∂F ( x, t ) = δ (1 − F ( x, t )) − ∂F ( x, t ) ∂t ∂x • Stationary: F ( x ) such that ∂F ( x,t ) = 0 . Integrating gives the ∂t 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 { e t } to maximize expected discounted utility: U ( c, ℓ ) = log c + β log ℓ c t = ψ t x t e t + ℓ t + τ = 1 dx t = µ ( e t ) x t dt + σx t dB t µ ( e ) = φe x = idiosyncratic productivity of a variety ψ t = 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: log ψ t + log x t + β log(Ω − e t ) + E [ dV ( x t , t )] ρV ( x t , t ) = max dt e t +( δ + ¯ δ )( V w ( t ) − V ( x t , t )) where Ω ≡ 1 − τ • Note: the “capital gain” term is E [ dV ( x t , t )] = µ ( e t ) x t V x ( x t , t ) + 1 2 σ 2 x 2 t V xx ( x t , t ) + V t ( x t , t ) dt 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 → x 0 . ◦ Endogenous creative destruction ( δ ): inherit existing productivity x . • Distribution f ( x, t ) satisfies Kolmogorov forward equation: 2 · ∂ 2 ∂f ( x, t ) δf ( x, t ) − ∂ ∂x [ µ ( e ∗ ) xf ( x, t )] + 1 � � = − ¯ σ 2 x 2 f ( x, t ) ∂x 2 ∂t • Stationary distribution lim t →∞ f ( x, t ) = f ( x ) solves ∂f ( x,t ) = 0 ∂t A Schumpeterian Model of Top Income Inequality – p. 22
• Guess that f ( · ) takes the Pareto form f ( x ) = Cx − ξ − 1 ⇒ �� ˜ � 2 + 2 ¯ ξ ∗ = − ˜ µ ∗ µ ∗ δ σ 2 + σ 2 σ 2 µ ∗ ≡ µ ( e ∗ ) − 1 δ ) − 1 2 σ 2 = φ (1 − τ ) − β ( ρ + δ ∗ + ¯ 2 σ 2 ˜ • Power-law inequality is therefore given by η ∗ = 1 /ξ ∗ A Schumpeterian Model of Top Income Inequality – p. 23
Comparative Statics (given δ ∗ ) �� ˜ � 2 + 2 ¯ ξ ∗ = − ˜ µ ∗ µ ∗ δ η ∗ = 1 /ξ ∗ , σ 2 + σ 2 σ 2 δ ) − 1 µ ∗ = φ (1 − τ ) − β ( ρ + δ ∗ + ¯ 2 σ 2 ˜ • 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 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: φ H > φ L with Poisson rate ¯ p of transition ( H → L ) A Schumpeterian Model of Top Income Inequality – p. 25
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