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% 6% United States 4% 2% France 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 14 12 10 8 6 Canada Ireland Switzerland Australia Italy Japan France Spain Norway New Zealand Sweden Mauritius Denmark Singapore 45 degree line 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% 8% Capital gains 6% 4% Business 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 6 Equals 1 1 η if Pareto... 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 Pr[Y > y] = ( y y 0 ) 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) = 10 η 1 A Schumpeterian Model of Top Income Inequality p. 8
Fractal Inequality Shares in the United States Fractal shares (percent) 45 40 35 From 20% in 1970 to 35% in 2010 S(.01) 30 25 20 S(.1) S(1) 15 1950 1960 1970 1980 1990 2000 2010 Year A Schumpeterian Model of Top Income Inequality p. 9
The Power-Law Inequality Exponentη, United States 1 + log 10 (top share) 0.65 0.6 0.55 0.5 η rises from.33 in 1970 to.55 in 2010 η(.01) 0.45 0.4 η(1) 0.35 η(.1) 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 If Pr[x i > x] = x 1/η x, then y i = wx α i Pr[y i > y] = ( ȳ w ) 1/ηy where η y = αη x 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 Exponential growth Creative destruction 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 thany? Equals fraction with at least x(y) years of experience Therefore x(y) = 1 µ log ( y y 0 ) Pr[Income > y] = Pr[Experience > x(y)] = e δx(y) ( y = y 0 ) δ µ 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)) }{{} inflow from above x Dividing both sides by t = x and taking the limit [F(x,t) F(x x,t)] }{{} outflow as top folks age F(x,t) t = δ(1 F(x,t)) F(x,t) x Stationary: F(x) such that F(x,t) exponential solution. t = 0. Integrating gives the 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) = logc+βlogl 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 = determined in GE (grows) δ = endogenous creative destruction δ = exogenous destruction ψ t A Schumpeterian Model of Top Income Inequality p. 19
Entrepreneur s Problem HJB Form The Bellman equation for the entreprenueur: ρv(x t,t) = max e t logψ t +logx t +βlog(ω e t )+ E[dV(x t,t)] dt +(δ + δ)(v w (t) V(x t,t)) where Ω 1 τ Note: the capital gain term is E[dV(x t,t)] dt = µ(e t )x t V x (x t,t)+ 1 2 σ2 x 2 tv xx (x t,t)+v t (x t,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 x 0. 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 [ σ 2 x 2 x 2 f(x,t) ] Stationary distribution lim t 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 σ 2 + ( µ σ 2 ) 2 + 2 δ σ 2 µ µ(e ) 1 2 σ2 = φ(1 τ) β(ρ+δ + δ) 1 2 σ2 Power-law inequality is therefore given by η = 1/ξ A Schumpeterian Model of Top Income Inequality p. 23
Comparative Statics (givenδ ) η = 1/ξ, ξ = µ σ 2 + ( µ σ 2 ) 2 + 2 δ σ 2 µ = φ(1 τ) β(ρ+δ + δ) 1 2 σ2 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
Pareto Inequality with Heterogeneous Growth Rates η = 1/ξ H, ξ H = µ H σ 2 + ) ( µ 2 H σ 2 + 2( δ + p) σ 2 µ H = φ H (1 τ) β(ρ+δ + δ) 1 2 σ2 This adopts Gabaix, Lasry, Lions, and Moll (2015) Why it helps quantitatively: φ H : 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 Y θ i di ) 1/θ Production of variety i Y i = γ n t xα i L i Resource constraint L t +R t +1 = N, L t 1 0 L itdi Flow rate of innovation ṅ t = λ(1 z)r t Creative destruction δ t = ṅ t A Schumpeterian Model of Top Income Inequality p. 27
Equilibrium with Monopolistic Competition Suppose R/ L = s where L N 1. Define X 1 0 x idi = x 0 1 η. Markup is 1/θ. Aggregate PF Y t = γ n t Xα L Wage for L Profits for variety i w t = θγ n t Xα π it = (1 θ)γ n t Xα L ( x i ) ( X xi ) wt X Definition of ψ t ψ t = (1 θ)γ n t Xα 1 L 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) s L g y = ṅlogγ = λ(1 z) s Llogγ 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. 29
Endogenizing Research and Growth A Schumpeterian Model of Top Income Inequality p. 30
Endogenizings = R/ L Worker: ρv w (t) = logw t + dv W (t) dt Researcher: ρv R (t) = log( mw t )+ dv R (t) dt + λ ( E[V(x,t)] V R (t) ) + δ R ( 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 = φ H (1 τ) β(ρ+δ + δ) 1 2 σ2 H Pareto inequality η = 1/ξ, ξ = µ H σ 2 H + ) ( µ 2 H σ + 2( δ+ p) H 2 σh 2 Creative destruction δ = λ(1 z)s L Growth g = δ logγ Research allocation V w (s ) = V R (s ) A Schumpeterian Model of Top Income Inequality p. 32
Varying the x-technology parameterφ POWER LAW INEQUALITY 1 GROWTH RATE (PERCENT) 4 0.75 3 0.50 2 0.25 1 0 0 0.3 0.4 0.5 0.6 0.7 A Schumpeterian Model of Top Income Inequality p. 33
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 x i X is more dispersed Elogπ i /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. 34
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. 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 in medium 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 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. 38
Micro Evidence A Schumpeterian Model of Top Income Inequality p. 39
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. 40
Growth Rates of Top 10% Incomes, 1995 1996 DENSITY 6 5 1 in 100: rise by a factor of 3.0 1 in 1,000: rise by a factor of 6.8 1 in 10,000: rise by a factor of 24.6 4 3 2 1 0 From Guvenen et al (2015) δ +δ { }} { µ H {}}{ -5-4 -3-2 -1 0 1 2 3 4 ANNUAL LOG CHANGE, 1995-96 A Schumpeterian Model of Top Income Inequality p. 41
Growth Rates of Top 5% Incomes, 1988 1989 Number of observations 180 160 140 120 100 80 60 40 20 0 4 3 2 1 0 1 2 Change in log income 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... σ H 0.122... p 0.767... µ H 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 structural parameters, 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 µ H.3 σ H = σ L =.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. 46
Numerical Example: Matching U.S. Inequality POWER LAW INEQUALITY 0.6 φ H in US rises from 0.385 to 0.568 GROWTH RATE (PERCENT) 3.0 0.5 2.5 0.4 0.3 US, η (left scale) US Growth (right scale) 2.00 1.5 0.2 1980 1985 1990 1995 2000 2005 1.0 A Schumpeterian Model of Top Income Inequality p. 47
Numerical Example: U.S. and France POWER LAW INEQUALITY 0.6 GROWTH RATE (PERCENT) 3.0 0.5 z in France falls from 0.350 to 0.250 p in France rises from 0.89 to 1.09 US, η 2.5 0.4 2.00 0.3 France, η 1.5 0.2 1980 1985 1990 1995 2000 2005 1.0 A Schumpeterian Model of Top Income Inequality p. 48
Numerical Example: Taxes and Inequality POWER LAW INEQUALITY 0.6 0.5 τ in the U.S. falls from 0.350 to 0.038 τ in France falls from 0.395 to 0.250 GROWTH RATE (PERCENT) 3.0 US, η 2.5 0.4 2.00 0.3 France, η 1.5 0.2 1980 1985 1990 1995 2000 2005 1.0 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: 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. 50