Entrepreneurial Taxation, Occupational Choice and
Credit Market Frictions
Florian Scheuer
MIT
May 2010
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
1 / 29
Introduction
Motivation
How should business profits be taxed relative to other forms of income?
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
2 / 29
Introduction
Motivation
How should business profits be taxed relative to other forms of income?
• Direct redistribution (tagging)
→ Entrepreneurs tend to be better off
→ Higher profit taxation
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
2 / 29
Introduction
Motivation
How should business profits be taxed relative to other forms of income?
• Direct redistribution (tagging)
→ Entrepreneurs tend to be better off
→ Higher profit taxation
• Indirect redistribution (trickle down)
→ Reducing business taxes encourages entrepreneurship, labor demand
→ Wages rise, benefits low to medium income workers
→ Subsidize entrepreneurial effort?
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
2 / 29
Introduction
Motivation
How should business profits be taxed relative to other forms of income?
• Direct redistribution (tagging)
→ Entrepreneurs tend to be better off
→ Higher profit taxation
• Indirect redistribution (trickle down)
→ Reducing business taxes encourages entrepreneurship, labor demand
→ Wages rise, benefits low to medium income workers
→ Subsidize entrepreneurial effort?
• Correct inefficiencies (entrepreneurs face credit market frictions)
→ Insufficient number of entrepreneurs?
→ Subsidize entry into entrepreneurship?
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
2 / 29
Introduction
Motivation
How should business profits be taxed relative to other forms of income?
• Direct redistribution (tagging)
→ Entrepreneurs tend to be better off
→ Higher profit taxation
• Indirect redistribution (trickle down)
→ Reducing business taxes encourages entrepreneurship, labor demand
→ Wages rise, benefits low to medium income workers
→ Subsidize entrepreneurial effort?
• Correct inefficiencies (entrepreneurs face credit market frictions)
→ Insufficient number of entrepreneurs?
→ Subsidize entry into entrepreneurship?
Key ingredients to model these tradeoffs:
• Heterogeneity in skill and occupational preference
• Endogenous occupational choice
• Entrepreneurs hire workers, endogenous wages
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
2 / 29
Introduction
Preview of Main Results
Baseline model without credit market frictions
• Uniform taxation, treating profits and labor income the same
→ Manipulate wages to achieve more redistribution (trickle down)
→ Production distortions
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
3 / 29
Introduction
Preview of Main Results
Baseline model without credit market frictions
• Uniform taxation, treating profits and labor income the same
→ Manipulate wages to achieve more redistribution (trickle down)
→ Production distortions
• Differential taxation of profits and labor income
→ Direct redistribution (tagging), production efficiency
→ Compare optimal profit and income tax schedules
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
3 / 29
Introduction
Preview of Main Results
Baseline model without credit market frictions
• Uniform taxation, treating profits and labor income the same
→ Manipulate wages to achieve more redistribution (trickle down)
→ Production distortions
• Differential taxation of profits and labor income
→ Direct redistribution (tagging), production efficiency
→ Compare optimal profit and income tax schedules
Add investment, borrowing and credit markets
• Adverse selection from private heterogeneity among entrepreneurs
• Credit market equilibrium affects entry into entrepreneurship
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
3 / 29
Introduction
Preview of Main Results
Baseline model without credit market frictions
• Uniform taxation, treating profits and labor income the same
→ Manipulate wages to achieve more redistribution (trickle down)
→ Production distortions
• Differential taxation of profits and labor income
→ Direct redistribution (tagging), production efficiency
→ Compare optimal profit and income tax schedules
Add investment, borrowing and credit markets
• Adverse selection from private heterogeneity among entrepreneurs
• Credit market equilibrium affects entry into entrepreneurship
→ Cross-subsidization from high to low-quality borrowers
→ Excessive (insufficient) entry of low-skill (high-skill) entrepreneurs
→ Regressive entrepreneurial taxation restores efficient occupational choice
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
3 / 29
Introduction
Related Literature
• Simulating tax reforms in quantitative models with entrepreneurs
Meh (2005) (flat tax reform), Cagetti/DeNardi (2009) (estate tax),
Kitao (2008), Panousi (2008) (capital tax)
• Optimal savings distortions with entrepreneurial investment
Albanesi (2006), (2008), Chari et al. (2002)
• Credit market interventions with adverse selection
Stiglitz/Weiss (1976), DeMeza/Webb (1987), Innes (1992), Parker (2003)
• Optimal taxation with endogenous wages
Feldstein (1973), Zeckhauser (1977), Allen (1982), Boadway et al. (1991),
Parker (1999), Stiglitz (1982), Moresi (1998), Naito (1999)
• Nonlinear taxation with multidimensional private information
Kleven et al. (2009)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
4 / 29
The Model
Baseline Model
Measure one of heterogeneous individuals with two-dimensional private type
(θ, φ) ∈ [θ, θ] × [0, φθ ],
Florian Scheuer (MIT)
cdf F (θ) and Gθ (φ)
Entrepreneurial Taxation
May 2010
5 / 29
The Model
Baseline Model
Measure one of heterogeneous individuals with two-dimensional private type
(θ, φ) ∈ [θ, θ] × [0, φθ ],
cdf F (θ) and Gθ (φ)
Individuals can choose between two occupations:
• Workers supply effective labor l at (endogenous) wage w
Quasi-linear preferences
U(c, l, θ) = c − ψ(l/θ),
Florian Scheuer (MIT)
ψ(.) increasing, convex
Entrepreneurial Taxation
May 2010
5 / 29
The Model
Baseline Model
Measure one of heterogeneous individuals with two-dimensional private type
(θ, φ) ∈ [θ, θ] × [0, φθ ],
cdf F (θ) and Gθ (φ)
Individuals can choose between two occupations:
• Workers supply effective labor l at (endogenous) wage w
Quasi-linear preferences
U(c, l, θ) = c − ψ(l/θ),
ψ(.) increasing, convex
• Entrepreneurs hire effective labor L and provide effective effort E
Profits
π = Y (L, E ) − wL, Y (L, E ) is CRS, concave
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
5 / 29
The Model
Baseline Model
Measure one of heterogeneous individuals with two-dimensional private type
(θ, φ) ∈ [θ, θ] × [0, φθ ],
cdf F (θ) and Gθ (φ)
Individuals can choose between two occupations:
• Workers supply effective labor l at (endogenous) wage w
Quasi-linear preferences
U(c, l, θ) = c − ψ(l/θ),
ψ(.) increasing, convex
• Entrepreneurs hire effective labor L and provide effective effort E
Profits
π = Y (L, E ) − wL, Y (L, E ) is CRS, concave
Utility
U(π, E , θ) = π − ψ(E /θ) − φ,
where φ is a (heterogeneous) cost of becoming an entrepreneur
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
5 / 29
No Tax Equilibrium
Equilibrium without Taxes I
Given w , conditional on becoming a worker, type θ solves maxl wl − ψ(l/θ)
→ l ∗ (θ, w ) and indirect utility vW (θ, w )
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
6 / 29
No Tax Equilibrium
Equilibrium without Taxes I
Given w , conditional on becoming a worker, type θ solves maxl wl − ψ(l/θ)
→ l ∗ (θ, w ) and indirect utility vW (θ, w )
If becoming an entrepreneur, type θ solves maxL,E Y (L, E ) − wL − ψ(E /θ)
→ L∗ (θ, w ), E ∗ (θ, w ) and indirect utility vE (θ, w )
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
6 / 29
No Tax Equilibrium
Equilibrium without Taxes I
Given w , conditional on becoming a worker, type θ solves maxl wl − ψ(l/θ)
→ l ∗ (θ, w ) and indirect utility vW (θ, w )
If becoming an entrepreneur, type θ solves maxL,E Y (L, E ) − wL − ψ(E /θ)
→ L∗ (θ, w ), E ∗ (θ, w ) and indirect utility vE (θ, w )
Critical cost value for occupational choice:
φ̃(θ, w ) ≡ vE (θ, w ) − vW (θ, w )
All (θ, φ) with φ ≤ φ̃(θ, w ) become entrepreneurs, the others workers
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
6 / 29
No Tax Equilibrium
Equilibrium without Taxes I
Given w , conditional on becoming a worker, type θ solves maxl wl − ψ(l/θ)
→ l ∗ (θ, w ) and indirect utility vW (θ, w )
If becoming an entrepreneur, type θ solves maxL,E Y (L, E ) − wL − ψ(E /θ)
→ L∗ (θ, w ), E ∗ (θ, w ) and indirect utility vE (θ, w )
Critical cost value for occupational choice:
φ̃(θ, w ) ≡ vE (θ, w ) − vW (θ, w )
All (θ, φ) with φ ≤ φ̃(θ, w ) become entrepreneurs, the others workers
Definition
An equilibrium without taxes is a wage w ∗ and an allocation {l ∗ (θ, w ∗ ),
L∗ (θ, w ∗ ), E ∗ (θ, w ∗ )} such that the labor market clears:
Z
Z
Gθ (φ̃(θ, w ∗ ))L∗ (θ, w ∗ )dF (θ) = (1 − Gθ (φ̃(θ, w ∗ )))l ∗ (θ, w ∗ )dF (θ)
Θ
Florian Scheuer (MIT)
Θ
Entrepreneurial Taxation
May 2010
6 / 29
No Tax Equilibrium
Equilibrium without Taxes II
Given E , w , entrepreneurs of all types θ solve
max Y (L, E ) − wL
L
⇒
YL (Lc (E , w ), E ) = w
With CRS,
Y (Lc (E , w ), E ) = YL (Lc (E , w ), E )Lc (E , w ) + YE (Lc (E , w ), E )E ,
and thus π = YE (Lc (E , w ), E )E
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
7 / 29
No Tax Equilibrium
Equilibrium without Taxes II
Given E , w , entrepreneurs of all types θ solve
max Y (L, E ) − wL
L
⇒
YL (Lc (E , w ), E ) = w
With CRS,
Y (Lc (E , w ), E ) = YL (Lc (E , w ), E )Lc (E , w ) + YE (Lc (E , w ), E )E ,
and thus π = YE (Lc (E , w ), E )E
→ Entrepreneurs can be thought of just receiving a different wage w̃ = YE
→ One-to-one relationship w̃ (w ), decreasing
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
7 / 29
No Tax Equilibrium
Equilibrium without Taxes II
Given E , w , entrepreneurs of all types θ solve
max Y (L, E ) − wL
L
⇒
YL (Lc (E , w ), E ) = w
With CRS,
Y (Lc (E , w ), E ) = YL (Lc (E , w ), E )Lc (E , w ) + YE (Lc (E , w ), E )E ,
and thus π = YE (Lc (E , w ), E )E
→ Entrepreneurs can be thought of just receiving a different wage w̃ = YE
→ One-to-one relationship w̃ (w ), decreasing
Lemma
(i) Any no tax equilibrium involves w̃ (w ∗ ) > w ∗ and E ∗ (θ, w̃ ∗ ) > l ∗ (θ, w ∗ ) ∀θ.
(ii) The critical cost value φ̃(θ, w ∗ ) is increasing in θ
(iii) The share of entrepreneurs G (φ̃(θ, w ∗ )) is increasing in θ if, for instance,
Gθ0 (φ) FOSD Gθ (φ) for θ0 ≤ θ
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
7 / 29
No Tax Equilibrium
Equilibrium without Taxes II
Given E , w , entrepreneurs of all types θ solve
max Y (L, E ) − wL
L
⇒
YL (Lc (E , w ), E ) = w
With CRS,
Y (Lc (E , w ), E ) = YL (Lc (E , w ), E )Lc (E , w ) + YE (Lc (E , w ), E )E ,
and thus π = YE (Lc (E , w ), E )E
→ Entrepreneurs can be thought of just receiving a different wage w̃ = YE
→ One-to-one relationship w̃ (w ), decreasing
Lemma
(i) Any no tax equilibrium involves w̃ (w ∗ ) > w ∗ and E ∗ (θ, w̃ ∗ ) > l ∗ (θ, w ∗ ) ∀θ.
(ii) The critical cost value φ̃(θ, w ∗ ) is increasing in θ
(iii) The share of entrepreneurs G (φ̃(θ, w ∗ )) is increasing in θ if, for instance,
Gθ0 (φ) FOSD Gθ (φ) for θ0 ≤ θ
Gθ (φ) can produce more general relationships between skill and entrepreneurship
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
7 / 29
Uniform Tax Treatment of Profits and Income
Uniform Taxation
Non-linear income tax T (.), treating labor income and firm profits equally
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
8 / 29
Uniform Tax Treatment of Profits and Income
Uniform Taxation
Non-linear income tax T (.), treating labor income and firm profits equally
Given E , w , entrepreneurs of all types θ solve
max Y (L, E ) − wL − T (Y (L, E ) − wL)
L
Florian Scheuer (MIT)
⇒
Entrepreneurial Taxation
YL = w
⇒
YE = w̃
May 2010
8 / 29
Uniform Tax Treatment of Profits and Income
Uniform Taxation
Non-linear income tax T (.), treating labor income and firm profits equally
Given E , w , entrepreneurs of all types θ solve
max Y (L, E ) − wL − T (Y (L, E ) − wL)
L
⇒
YL = w
⇒
YE = w̃
Entrepreneur of type θ solves maxE w̃ E − T (w̃ E ) − ψ(E /θ)
Worker of type θ solves maxl wl − T (wl) − ψ(l/θ)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
8 / 29
Uniform Tax Treatment of Profits and Income
Uniform Taxation
Non-linear income tax T (.), treating labor income and firm profits equally
Given E , w , entrepreneurs of all types θ solve
max Y (L, E ) − wL − T (Y (L, E ) − wL)
L
⇒
YL = w
⇒
YE = w̃
Entrepreneur of type θ solves maxE w̃ E − T (w̃ E ) − ψ(E /θ)
Worker of type θ solves maxl wl − T (wl) − ψ(l/θ)
→ Entrepreneur of type θ and worker of type θ0 s.t. w̃ θ = w θ0 choose
w̃
w̃
w̃ E (θ) = wl
θ
⇒ vE (θ) = vW
θ
w
w
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
8 / 29
Uniform Tax Treatment of Profits and Income
Uniform Taxation
Non-linear income tax T (.), treating labor income and firm profits equally
Given E , w , entrepreneurs of all types θ solve
max Y (L, E ) − wL − T (Y (L, E ) − wL)
L
⇒
YL = w
⇒
YE = w̃
Entrepreneur of type θ solves maxE w̃ E − T (w̃ E ) − ψ(E /θ)
Worker of type θ solves maxl wl − T (wl) − ψ(l/θ)
→ Entrepreneur of type θ and worker of type θ0 s.t. w̃ θ = w θ0 choose
w̃
w̃
w̃ E (θ) = wl
θ
⇒ vE (θ) = vW
θ
w
w
→ Cannot discriminate entrepreneurs of skill θ and workers of skill (w̃ /w )θ
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
8 / 29
Uniform Tax Treatment of Profits and Income
Constrained Pareto Problem
With Pareto-weights F̃ (θ), G̃θ (φ), the constrained Pareto problem is
Uniform Taxation
max
E (θ), L(θ), l(θ),
vE (θ), vW (θ), w , w̃
Z h
Z Z
i
G̃θ (φ̃(θ))vE (θ) + (1−G̃θ (φ̃(θ)))vw (θ) d F̃ (θ)−
Θ
Θ
Florian Scheuer (MIT)
Entrepreneurial Taxation
φ̃(θ)
φd G̃θ (φ)d F̃ (θ)
φ
May 2010
9 / 29
Uniform Tax Treatment of Profits and Income
Constrained Pareto Problem
With Pareto-weights F̃ (θ), G̃θ (φ), the constrained Pareto problem is
Uniform Taxation
max
E (θ), L(θ), l(θ),
vE (θ), vW (θ), w , w̃
Z h
Z Z
i
G̃θ (φ̃(θ))vE (θ) + (1−G̃θ (φ̃(θ)))vw (θ) d F̃ (θ)−
Θ
Θ
s.t.
Florian Scheuer (MIT)
φ̃(θ)
φd G̃θ (φ)d F̃ (θ)
φ
φ̃(θ) = vE (θ) − vW (θ) ∀θ ∈ Θ
Entrepreneurial Taxation
May 2010
9 / 29
Uniform Tax Treatment of Profits and Income
Constrained Pareto Problem
With Pareto-weights F̃ (θ), G̃θ (φ), the constrained Pareto problem is
Uniform Taxation
max
E (θ), L(θ), l(θ),
vE (θ), vW (θ), w , w̃
Z h
Z Z
i
G̃θ (φ̃(θ))vE (θ) + (1−G̃θ (φ̃(θ)))vw (θ) d F̃ (θ)−
Θ
Θ
s.t.
φ̃(θ)
φd G̃θ (φ)d F̃ (θ)
φ
φ̃(θ) = vE (θ) − vW (θ) ∀θ ∈ Θ
vE (θ)≥vE (θ0 )+ψ(E (θ0 )/θ0 )−ψ(E (θ0 )/θ), vW (θ)≥vW (θ0 )+ψ(l(θ0 )/θ0 )−ψ(l(θ0 )/θ) ∀θ,θ0∈Θ
(IC)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
9 / 29
Uniform Tax Treatment of Profits and Income
Constrained Pareto Problem
With Pareto-weights F̃ (θ), G̃θ (φ), the constrained Pareto problem is
Uniform Taxation
max
E (θ), L(θ), l(θ),
vE (θ), vW (θ), w , w̃
Z h
Z Z
i
G̃θ (φ̃(θ))vE (θ) + (1−G̃θ (φ̃(θ)))vw (θ) d F̃ (θ)−
Θ
Θ
s.t.
φ̃(θ)
φd G̃θ (φ)d F̃ (θ)
φ
φ̃(θ) = vE (θ) − vW (θ) ∀θ ∈ Θ
vE (θ)≥vE (θ0 )+ψ(E (θ0 )/θ0 )−ψ(E (θ0 )/θ), vW (θ)≥vW (θ0 )+ψ(l(θ0 )/θ0 )−ψ(l(θ0 )/θ) ∀θ,θ0∈Θ
(IC)
Z
Z
Gθ (φ̃(θ))L(θ)dF (θ) ≤
Θ
Florian Scheuer (MIT)
(1 − Gθ (φ̃(θ)))l(θ)dF (θ)
(LM)
Θ
Entrepreneurial Taxation
May 2010
9 / 29
Uniform Tax Treatment of Profits and Income
Constrained Pareto Problem
With Pareto-weights F̃ (θ), G̃θ (φ), the constrained Pareto problem is
Uniform Taxation
Z h
max
E (θ), L(θ), l(θ),
vE (θ), vW (θ), w , w̃
Z Z
i
G̃θ (φ̃(θ))vE (θ) + (1−G̃θ (φ̃(θ)))vw (θ) d F̃ (θ)−
Θ
Θ
s.t.
φ̃(θ)
φd G̃θ (φ)d F̃ (θ)
φ
φ̃(θ) = vE (θ) − vW (θ) ∀θ ∈ Θ
vE (θ)≥vE (θ0 )+ψ(E (θ0 )/θ0 )−ψ(E (θ0 )/θ), vW (θ)≥vW (θ0 )+ψ(l(θ0 )/θ0 )−ψ(l(θ0 )/θ) ∀θ,θ0∈Θ
(IC)
Z
Z
Gθ (φ̃(θ))L(θ)dF (θ) ≤
Θ
(1 − Gθ (φ̃(θ)))l(θ)dF (θ)
(LM)
Θ
Z
Θ
Gθ (φ̃(θ)) [Y (L(θ), E (θ)) − vE (θ) − ψ(E (θ)/θ)] dF (θ)
Z
− (1 − G (φ̃(θ))) [vw (θ) + ψ(l(θ)/θ)] dF (θ) ≥
0
(RC)
Θ
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
9 / 29
Uniform Tax Treatment of Profits and Income
Constrained Pareto Problem
With Pareto-weights F̃ (θ), G̃θ (φ), the constrained Pareto problem is
Uniform Taxation
Z h
max
E (θ), L(θ), l(θ),
vE (θ), vW (θ), w , w̃
Z Z
i
G̃θ (φ̃(θ))vE (θ) + (1−G̃θ (φ̃(θ)))vw (θ) d F̃ (θ)−
Θ
Θ
s.t.
φ̃(θ)
φd G̃θ (φ)d F̃ (θ)
φ
φ̃(θ) = vE (θ) − vW (θ) ∀θ ∈ Θ
vE (θ)≥vE (θ0 )+ψ(E (θ0 )/θ0 )−ψ(E (θ0 )/θ), vW (θ)≥vW (θ0 )+ψ(l(θ0 )/θ0 )−ψ(l(θ0 )/θ) ∀θ,θ0∈Θ
(IC)
Z
Z
Gθ (φ̃(θ))L(θ)dF (θ) ≤
Θ
(1 − Gθ (φ̃(θ)))l(θ)dF (θ)
(LM)
Θ
Z
Θ
Gθ (φ̃(θ)) [Y (L(θ), E (θ)) − vE (θ) − ψ(E (θ)/θ)] dF (θ)
Z
− (1 − G (φ̃(θ))) [vw (θ) + ψ(l(θ)/θ)] dF (θ) ≥
0
(RC)
Θ
vE (θ) = vW ((w̃ /w )θ) ,
Florian Scheuer (MIT)
E (θ) = (w /w̃ )l ((w̃ /w )θ) ∀θ ∈ θ, (w /w̃ )θ
Entrepreneurial Taxation
May 2010
(ND)
9 / 29
Uniform Tax Treatment of Profits and Income
Constrained Pareto Problem
With Pareto-weights F̃ (θ), G̃θ (φ), the constrained Pareto problem is
Uniform Taxation
Z h
max
E (θ), L(θ), l(θ),
vE (θ), vW (θ), w , w̃
φ̃(θ)
Z Z
i
G̃θ (φ̃(θ))vE (θ) + (1−G̃θ (φ̃(θ)))vw (θ) d F̃ (θ)−
Θ
Θ
s.t.
φd G̃θ (φ)d F̃ (θ)
φ
φ̃(θ) = vE (θ) − vW (θ) ∀θ ∈ Θ
vE (θ)≥vE (θ0 )+ψ(E (θ0 )/θ0 )−ψ(E (θ0 )/θ), vW (θ)≥vW (θ0 )+ψ(l(θ0 )/θ0 )−ψ(l(θ0 )/θ) ∀θ,θ0∈Θ
(IC)
Z
Z
Gθ (φ̃(θ))L(θ)dF (θ) ≤
Θ
(1 − Gθ (φ̃(θ)))l(θ)dF (θ)
(LM)
Θ
Z
Θ
Gθ (φ̃(θ)) [Y (L(θ), E (θ)) − vE (θ) − ψ(E (θ)/θ)] dF (θ)
Z
− (1 − G (φ̃(θ))) [vw (θ) + ψ(l(θ)/θ)] dF (θ) ≥
0
(RC)
Θ
vE (θ) = vW ((w̃ /w )θ) ,
E (θ) = (w /w̃ )l ((w̃ /w )θ) ∀θ ∈ θ, (w /w̃ )θ
w = YL (L(θ), E (θ)),
Florian Scheuer (MIT)
w̃ = YE (L(θ), E (θ)) ∀θ ∈ Θ
Entrepreneurial Taxation
(ND)
(MP)
May 2010
9 / 29
Uniform Tax Treatment of Profits and Income
Constrained Pareto Problem
With Pareto-weights F̃ (θ), G̃θ (φ), the constrained Pareto problem is
Uniform Taxation
Z h
max
E (θ), L(θ), l(θ),
vE (θ), vW (θ), w , w̃
φ̃(θ)
Z Z
i
G̃θ (φ̃(θ))vE (θ) + (1−G̃θ (φ̃(θ)))vw (θ) d F̃ (θ)−
Θ
Θ
s.t.
φd G̃θ (φ)d F̃ (θ)
φ
φ̃(θ) = vE (θ) − vW (θ) ∀θ ∈ Θ
0
vE0 (θ) = E (θ)ψ 0 (E (θ)/θ) /θ2 , vW
(θ) = l(θ)ψ 0 (l(θ)/θ) /θ2 , E (θ), l(θ) increasing ∀θ ∈ Θ
(IC)
Z
Z
Gθ (φ̃(θ))L(θ)dF (θ) ≤
Θ
(1 − Gθ (φ̃(θ)))l(θ)dF (θ)
(LM)
Θ
Z
Θ
Gθ (φ̃(θ)) [Y (L(θ), E (θ)) − vE (θ) − ψ(E (θ)/θ)] dF (θ)
Z
− (1 − G (φ̃(θ))) [vw (θ) + ψ(l(θ)/θ)] dF (θ) ≥
0
(RC)
Θ
vE (θ) = vW ((w̃ /w )θ) ,
E (θ) = (w /w̃ )l ((w̃ /w )θ) ∀θ ∈ θ, (w /w̃ )θ
w = YL (L(θ), E (θ)),
Florian Scheuer (MIT)
w̃ = YE (L(θ), E (θ)) ∀θ ∈ Θ
Entrepreneurial Taxation
(ND)
(MP)
May 2010
10 / 29
Uniform Tax Treatment of Profits and Income
Optimal Taxation Results
Pecuniary externality from prices w and w̃ entering program through (ND)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
11 / 29
Uniform Tax Treatment of Profits and Income
Optimal Taxation Results
Pecuniary externality from prices w and w̃ entering program through (ND)
Proposition
(i) In any constrained Pareto optimum, w̃ > w and w̃ E (θ) > wl(θ) ∀θ ∈ Θ
(ii) T 0 (wl(θ)) = T 0 (w̃ E (θ)) = 0 if Y (L, E ) is linear
(iii) Otherwise, T 0 (wl(θ)) and T 0 (w̃ E (θ)) have opposite signs.
(iv) E.g. if G̃ FOSD G and F̃ = F , then T 0 (wl(θ)) > 0 and T 0 (w̃ E (θ)) < 0
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
11 / 29
Uniform Tax Treatment of Profits and Income
Optimal Taxation Results
Pecuniary externality from prices w and w̃ entering program through (ND)
Proposition
(i) In any constrained Pareto optimum, w̃ > w and w̃ E (θ) > wl(θ) ∀θ ∈ Θ
(ii) T 0 (wl(θ)) = T 0 (w̃ E (θ)) = 0 if Y (L, E ) is linear
(iii) Otherwise, T 0 (wl(θ)) and T 0 (w̃ E (θ)) have opposite signs.
(iv) E.g. if G̃ FOSD G and F̃ = F , then T 0 (wl(θ)) > 0 and T 0 (w̃ E (θ)) < 0
Affect wages to relax (ND) and achieve more redistribution (trickle down)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
11 / 29
Uniform Tax Treatment of Profits and Income
Optimal Taxation Results
Pecuniary externality from prices w and w̃ entering program through (ND)
Proposition
(i) In any constrained Pareto optimum, w̃ > w and w̃ E (θ) > wl(θ) ∀θ ∈ Θ
(ii) T 0 (wl(θ)) = T 0 (w̃ E (θ)) = 0 if Y (L, E ) is linear
(iii) Otherwise, T 0 (wl(θ)) and T 0 (w̃ E (θ)) have opposite signs.
(iv) E.g. if G̃ FOSD G and F̃ = F , then T 0 (wl(θ)) > 0 and T 0 (w̃ E (θ)) < 0
Affect wages to relax (ND) and achieve more redistribution (trickle down)
Proposition
If the gov’t could distort YL (L(θ), E (θ)) across firms, then it would in general be
optimal to do so whenever Y (L, E ) is not linear and (ND) binds for some θ
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
11 / 29
Uniform Tax Treatment of Profits and Income
Optimal Taxation Results
Pecuniary externality from prices w and w̃ entering program through (ND)
Proposition
(i) In any constrained Pareto optimum, w̃ > w and w̃ E (θ) > wl(θ) ∀θ ∈ Θ
(ii) T 0 (wl(θ)) = T 0 (w̃ E (θ)) = 0 if Y (L, E ) is linear
(iii) Otherwise, T 0 (wl(θ)) and T 0 (w̃ E (θ)) have opposite signs.
(iv) E.g. if G̃ FOSD G and F̃ = F , then T 0 (wl(θ)) > 0 and T 0 (w̃ E (θ)) < 0
Affect wages to relax (ND) and achieve more redistribution (trickle down)
Proposition
If the gov’t could distort YL (L(θ), E (θ)) across firms, then it would in general be
optimal to do so whenever Y (L, E ) is not linear and (ND) binds for some θ
Production distortions to relax (ND)
Production efficiency (Diamond/Mirrlees, 1971) not generally optimal
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
11 / 29
Differential Tax Treatment of Profits and Income
Differential Income and Profit Taxation
Allow different tax schedules Ty (.) for labor income y ≡ wl and Tπ (.) for profits π
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
12 / 29
Differential Tax Treatment of Profits and Income
Differential Income and Profit Taxation
Allow different tax schedules Ty (.) for labor income y ≡ wl and Tπ (.) for profits π
Relaxed constrained Pareto problem:
Differential Taxation
Z h
max
E (θ), L(θ), l(θ),
vE (θ), vW (θ)
Z Z
i
G̃θ (φ̃(θ))vE (θ) + (1−G̃θ (φ̃(θ)))vw (θ) d F̃ (θ)−
Θ
Θ
s.t.
φ̃(θ)
φd G̃θ (φ)d F̃ (θ)
φ
φ̃(θ) = vE (θ) − vW (θ) ∀θ ∈ Θ
0
vE0 (θ) = E (θ)ψ 0 (E (θ)/θ) /θ2 , vW
(θ) = l(θ)ψ 0 (l(θ)/θ) /θ2 , E (θ), l(θ) increasing ∀θ ∈ Θ
(IC)
Z
Z
Gθ (φ̃(θ))L(θ)dF (θ) ≤
Θ
(1 − Gθ (φ̃(θ)))l(θ)dF (θ)
(LM)
Θ
Z
Θ
Gθ (φ̃(θ)) [Y (L(θ), E (θ)) − vE (θ) − ψ(E (θ)/θ)] dF (θ)
Z
− (1 − G (φ̃(θ))) [vw (θ) + ψ(l(θ)/θ)] dF (θ) ≥
0
(RC)
Θ
Same as with uniform taxation, but without constraints (ND) and (MP)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
12 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results I
Differential taxation eliminates pecuniary externalities
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
13 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results I
Differential taxation eliminates pecuniary externalities
Proposition
(i) At any optimum, YL (L(θ), E (θ)) is equalized across all θ ∈ Θ
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
13 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results I
Differential taxation eliminates pecuniary externalities
Proposition
(i) At any optimum, YL (L(θ), E (θ)) is equalized across all θ ∈ Θ
(ii) If there is no bunching, Tπ0 (π(θ)) and Ty0 (y (θ)) satisfy
Z
i
Tπ0 (π(θ))
1 + 1/ε(π(θ)) θh
=
G̃θ̂ (φ̃(θ̂))f˜(θ̂)−Gθ̂ (φ̃(θ̂))f (θ̂) +gθ̂ (φ̃(θ̂))∆T (θ̂)f (θ̂) d θ̂
0
1 − Tπ (π(θ)) θf (θ)Gθ (φ̃(θ)) θ
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
13 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results I
Differential taxation eliminates pecuniary externalities
Proposition
(i) At any optimum, YL (L(θ), E (θ)) is equalized across all θ ∈ Θ
(ii) If there is no bunching, Tπ0 (π(θ)) and Ty0 (y (θ)) satisfy
Z
i
Tπ0 (π(θ))
1 + 1/ε(π(θ)) θh
=
G̃θ̂ (φ̃(θ̂))f˜(θ̂)−Gθ̂ (φ̃(θ̂))f (θ̂) +gθ̂ (φ̃(θ̂))∆T (θ̂)f (θ̂) d θ̂
0
1 − Tπ (π(θ)) θf (θ)Gθ (φ̃(θ)) θ
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
13 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results I
Differential taxation eliminates pecuniary externalities
Proposition
(i) At any optimum, YL (L(θ), E (θ)) is equalized across all θ ∈ Θ
(ii) If there is no bunching, Tπ0 (π(θ)) and Ty0 (y (θ)) satisfy
Z
i
Tπ0 (π(θ))
1 + 1/ε(π(θ)) θh
=
G̃θ̂ (φ̃(θ̂))f˜(θ̂)−Gθ̂ (φ̃(θ̂))f (θ̂) +gθ̂ (φ̃(θ̂))∆T (θ̂)f (θ̂) d θ̂
0
1 − Tπ (π(θ)) θf (θ)Gθ (φ̃(θ)) θ
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
13 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results I
Differential taxation eliminates pecuniary externalities
Proposition
(i) At any optimum, YL (L(θ), E (θ)) is equalized across all θ ∈ Θ
(ii) If there is no bunching, Tπ0 (π(θ)) and Ty0 (y (θ)) satisfy
Z
i
Tπ0 (π(θ))
1 + 1/ε(π(θ)) θh
=
G̃θ̂ (φ̃(θ̂))f˜(θ̂)−Gθ̂ (φ̃(θ̂))f (θ̂) +gθ̂ (φ̃(θ̂))∆T (θ̂)f (θ̂) d θ̂
0
1 − Tπ (π(θ)) θf (θ)Gθ (φ̃(θ)) θ
Z θh
i
Ty0 (y (θ))
1 + 1/ε(y(θ))
(1−G̃θ̂(φ̃(θ̂)))f˜(θ̂)−(1−Gθ̂(φ̃(θ̂)))f (θ̂)−gθ̂(φ̃(θ̂))∆T(θ̂)f (θ̂) dθ̂
=
0
1−Ty (y(θ)) θf (θ)(1−Gθ(φ̃(θ))) θ
with ∆T (θ) ≡ Tπ (π(θ)) − Ty (y (θ))
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
13 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results I
Differential taxation eliminates pecuniary externalities
Proposition
(i) At any optimum, YL (L(θ), E (θ)) is equalized across all θ ∈ Θ
(ii) If there is no bunching, Tπ0 (π(θ)) and Ty0 (y (θ)) satisfy
Z
i
Tπ0 (π(θ))
1 + 1/ε(π(θ)) θh
=
G̃θ̂ (φ̃(θ̂))f˜(θ̂)−Gθ̂ (φ̃(θ̂))f (θ̂) +gθ̂ (φ̃(θ̂))∆T (θ̂)f (θ̂) d θ̂
0
1 − Tπ (π(θ)) θf (θ)Gθ (φ̃(θ)) θ
Z θh
i
Ty0 (y (θ))
1 + 1/ε(y(θ))
(1−G̃θ̂(φ̃(θ̂)))f˜(θ̂)−(1−Gθ̂(φ̃(θ̂)))f (θ̂)−gθ̂(φ̃(θ̂))∆T(θ̂)f (θ̂) dθ̂
=
0
1−Ty (y(θ)) θf (θ)(1−Gθ(φ̃(θ))) θ
with ∆T (θ) ≡ Tπ (π(θ)) − Ty (y (θ))
Observations:
• Production efficiency always optimal
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
13 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results I
Differential taxation eliminates pecuniary externalities
Proposition
(i) At any optimum, YL (L(θ), E (θ)) is equalized across all θ ∈ Θ
(ii) If there is no bunching, Tπ0 (π(θ)) and Ty0 (y (θ)) satisfy
Z
i
Tπ0 (π(θ))
1 + 1/ε(π(θ)) θh
=
G̃θ̂ (φ̃(θ̂))f˜(θ̂)−Gθ̂ (φ̃(θ̂))f (θ̂) +gθ̂ (φ̃(θ̂))∆T (θ̂)f (θ̂) d θ̂
0
1 − Tπ (π(θ)) θf (θ)Gθ (φ̃(θ)) θ
Z θh
i
Ty0 (y (θ))
1 + 1/ε(y(θ))
(1−G̃θ̂(φ̃(θ̂)))f˜(θ̂)−(1−Gθ̂(φ̃(θ̂)))f (θ̂)−gθ̂(φ̃(θ̂))∆T(θ̂)f (θ̂) dθ̂
=
0
1−Ty (y(θ)) θf (θ)(1−Gθ(φ̃(θ))) θ
with ∆T (θ) ≡ Tπ (π(θ)) − Ty (y (θ))
Observations:
• Production efficiency always optimal
• Optimal tax formulas no longer depend on whether Y (L, E ) is linear or not
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
13 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results I
Differential taxation eliminates pecuniary externalities
Proposition
(i) At any optimum, YL (L(θ), E (θ)) is equalized across all θ ∈ Θ
(ii) If there is no bunching, Tπ0 (π(θ)) and Ty0 (y (θ)) satisfy
Z
i
Tπ0 (π(θ))
1 + 1/ε(π(θ)) θh
=
G̃θ̂ (φ̃(θ̂))f˜(θ̂)−Gθ̂ (φ̃(θ̂))f (θ̂) +gθ̂ (φ̃(θ̂))∆T (θ̂)f (θ̂) d θ̂
0
1 − Tπ (π(θ)) θf (θ)Gθ (φ̃(θ)) θ
Z θh
i
Ty0 (y (θ))
1 + 1/ε(y(θ))
(1−G̃θ̂(φ̃(θ̂)))f˜(θ̂)−(1−Gθ̂(φ̃(θ̂)))f (θ̂)−gθ̂(φ̃(θ̂))∆T(θ̂)f (θ̂) dθ̂
=
0
1−Ty (y(θ)) θf (θ)(1−Gθ(φ̃(θ))) θ
with ∆T (θ) ≡ Tπ (π(θ)) − Ty (y (θ))
Observations:
• Production efficiency always optimal
• Optimal tax formulas no longer depend on whether Y (L, E ) is linear or not
• In particular, Tπ0 (π(θ)) = Tπ0 (π(θ)) = Ty0 (y (θ)) = Ty0 (y (θ)) = 0 in any case
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
13 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results II
Corollary
With a constant elasticity ε, the average marginal tax across occupations satisfies
T 0 (y (θ))
Tπ0 (π(θ))
1 + 1/ε y
+ 1 − Gθ (φ̃)
F̃ (θ) − F (θ)
Gθ (φ̃(θ))
=
0
0
1 − Tπ (π(θ))
1 − Ty (y (θ))
θf (θ)
Average tax in closed form and determined by redistribution across skills only
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
14 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results II
Corollary
With a constant elasticity ε, the average marginal tax across occupations satisfies
T 0 (y (θ))
Tπ0 (π(θ))
1 + 1/ε y
+ 1 − Gθ (φ̃)
F̃ (θ) − F (θ)
Gθ (φ̃(θ))
=
0
0
1 − Tπ (π(θ))
1 − Ty (y (θ))
θf (θ)
Average tax in closed form and determined by redistribution across skills only
Reintepretation: testing for Pareto optimality of Tπ , Ty
Corollary
The schedules Tπ , Ty inducing an allocation (π(θ), y (θ), φ̃(θ)) are Pareto optimal iff
Z θ
gθ̂ (φ̃(θ̂))f (θ̂)
θfE (θ)
Tπ0 (π(θ))
+
F
(θ)
−
∆T (θ̂)d θ̂ and
E
1 + 1/ε 1 − Tπ0 (π(θ))
G
θ
Z θ
Ty0 (y (θ))
gθ̂ (φ̃(θ̂))f (θ̂)
θfW (θ)
+ FW (θ) +
∆T (θ̂)d θ̂
0
1 + 1/ε 1 − Ty (y (θ))
1−G
θ
R
are increasing in θ, where G ≡ Θ Gθ (φ̃(θ))dF (θ) and
fE (θ) ≡ Gθ (φ̃(θ))f (θ)/G , fW (θ) ≡ (1 − Gθ (φ̃(θ)))f (θ)/(1 − G )
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
14 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results II
Corollary
With a constant elasticity ε, the average marginal tax across occupations satisfies
T 0 (y (θ))
Tπ0 (π(θ))
1 + 1/ε y
+ 1 − Gθ (φ̃)
F̃ (θ) − F (θ)
Gθ (φ̃(θ))
=
0
0
1 − Tπ (π(θ))
1 − Ty (y (θ))
θf (θ)
Average tax in closed form and determined by redistribution across skills only
Reintepretation: testing for Pareto optimality of Tπ , Ty
Corollary
The schedules Tπ , Ty inducing an allocation (π(θ), y (θ), φ̃(θ)) are Pareto optimal iff
Z θ
gθ̂ (φ̃(θ̂))f (θ̂)
θfE (θ)
Tπ0 (π(θ))
+
F
(θ)
−
∆T (θ̂)d θ̂ and
E
1 + 1/ε 1 − Tπ0 (π(θ))
G
θ
Z θ
Ty0 (y (θ))
gθ̂ (φ̃(θ̂))f (θ̂)
θfW (θ)
+ FW (θ) +
∆T (θ̂)d θ̂
0
1 + 1/ε 1 − Ty (y (θ))
1−G
θ
R
are increasing in θ, where G ≡ Θ Gθ (φ̃(θ))dF (θ) and
fE (θ) ≡ Gθ (φ̃(θ))f (θ)/G , fW (θ) ≡ (1 − Gθ (φ̃(θ)))f (θ)/(1 − G )
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
14 / 29
Differential Tax Treatment of Profits and Income
Optimal Taxation Results II
Corollary
The schedules Tπ , Ty inducing an allocation (π(θ), y (θ), φ̃(θ)) are Pareto optimal if
T 0 (y (θ)) Tπ0 (π(θ))
θf (θ)
y
Gθ (φ̃(θ))
+ F (θ) is increasing
+
1
−
G
(
φ̃)
θ
1 + 1/ε
1 − Tπ0 (π(θ))
1 − Ty0 (y (θ))
Only identification of F (θ) required
Reintepretation: testing for Pareto optimality of Tπ , Ty
Corollary
The schedules Tπ , Ty inducing an allocation (π(θ), y (θ), φ̃(θ)) are Pareto optimal iff
Z θ
gθ̂ (φ̃(θ̂))f (θ̂)
θfE (θ)
Tπ0 (π(θ))
+
F
(θ)−
∆T (θ̂)d θ̂ and
E
1 + 1/ε 1 − Tπ0 (π(θ))
G
θ
Z θ
Ty0 (y (θ))
gθ̂ (φ̃(θ̂))f (θ̂)
θfW (θ)
+ FW (θ)+
∆T (θ̂)d θ̂
0
1 + 1/ε 1 − Ty (y (θ))
1−G
θ
R
are increasing in θ, where G ≡ Θ Gθ (φ̃(θ))dF (θ) and
fE (θ) ≡ Gθ (φ̃(θ))f (θ)/G , fW (θ) ≡ (1 − Gθ (φ̃(θ)))f (θ)/(1 − G )
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
15 / 29
Differential Tax Treatment of Profits and Income
Comparing Profit and Income Taxes
Assumption 1
θ and φ are independent and g (φ) is non-increasing
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
16 / 29
Differential Tax Treatment of Profits and Income
Comparing Profit and Income Taxes
Assumption 1
θ and φ are independent and g (φ) is non-increasing
Redistribution across cost types
Proposition
Suppose F̃ (θ) = F (θ) and g̃ (φ) < g (φ) for all φ ≤ φ̃(θ). Then
(i) Ty0 (y (θ)) < 0, Tπ0 (π(θ)) > 0 for all θ ∈ (θ, θ)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
16 / 29
Differential Tax Treatment of Profits and Income
Comparing Profit and Income Taxes
Assumption 1
θ and φ are independent and g (φ) is non-increasing
Redistribution across cost types
Proposition
Suppose F̃ (θ) = F (θ) and g̃ (φ) < g (φ) for all φ ≤ φ̃(θ). Then
(i) Ty0 (y (θ)) < 0, Tπ0 (π(θ)) > 0 for all θ ∈ (θ, θ)
(ii) ∆T (θ) > 0 and ∆T 0 (θ) > 0 for all θ ∈ Θ
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
16 / 29
Differential Tax Treatment of Profits and Income
Comparing Profit and Income Taxes
Assumption 1
θ and φ are independent and g (φ) is non-increasing
Redistribution across cost types
Proposition
Suppose F̃ (θ) = F (θ) and g̃ (φ) < g (φ) for all φ ≤ φ̃(θ). Then
(i) Ty0 (y (θ)) < 0, Tπ0 (π(θ)) > 0 for all θ ∈ (θ, θ)
(ii) ∆T (θ) > 0 and ∆T 0 (θ) > 0 for all θ ∈ Θ
(iii) Compared with the no tax equilibrium, w = YL ↓, w̃ = YE ↑, φ̃(θ) ↓, L(θ) ↑
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
16 / 29
Differential Tax Treatment of Profits and Income
Comparing Profit and Income Taxes
Assumption 1
θ and φ are independent and g (φ) is non-increasing
Redistribution across cost types
Proposition
Suppose F̃ (θ) = F (θ) and g̃ (φ) < g (φ) for all φ ≤ φ̃(θ). Then
(i) Ty0 (y (θ)) < 0, Tπ0 (π(θ)) > 0 for all θ ∈ (θ, θ)
(ii) ∆T (θ) > 0 and ∆T 0 (θ) > 0 for all θ ∈ Θ
(iii) Compared with the no tax equilibrium, w = YL ↓, w̃ = YE ↑, φ̃(θ) ↓, L(θ) ↑
Redistribution across skill types
Proposition
Suppose that G̃ (φ) = G (φ) but F̃ (θ) 6= F (θ). If occupations are fixed, then
Ty0 (y (θ))
Tπ0 (π(θ))
1 + 1/ε =
=
F̃ (θ) − F (θ) for any w , w̃
0
0
1 − Tπ (π(θ))
1 − Ty (y (θ))
θf (θ)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
16 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Calibration I
Data on profits, income and entrepreneurship from 2007 SCF
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
17 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Calibration I
Data on profits, income and entrepreneurship from 2007 SCF
Entrepreneur: (i) self-employed, (ii) own business, (iii) actively manage it,
(iv) ≥ 2 employees
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
17 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Calibration I
Data on profits, income and entrepreneurship from 2007 SCF
Entrepreneur: (i) self-employed, (ii) own business, (iii) actively manage it,
(iv) ≥ 2 employees
Descriptive Statistics
Entrepreneurs
Mean
St. Dev.
Age
48.4
10.2
Yearly Income (in 1000$)
88.5
234.7
Hours per Week
48.3
14.1
50.2
6.0
Weeks per Year
Wage per Hour (in $)
55.5
243.8
Florian Scheuer (MIT)
Entrepreneurial Taxation
Workers
Mean St. Dev.
42.1
11.6
69.5
128.3
43.4
10.5
50.4
5.7
34.6
124.9
May 2010
17 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Calibration II
Constant elasticity disutility of effort ψ(e) = e 1+1/ε /(1 + 1/ε) with ε = .25
Cobb-Douglas technology Y (L, E ) = Lα E 1−α with α = .63 (workers’ share of
income in SCF data)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
18 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Calibration II
Constant elasticity disutility of effort ψ(e) = e 1+1/ε /(1 + 1/ε) with ε = .25
Cobb-Douglas technology Y (L, E ) = Lα E 1−α with α = .63 (workers’ share of
income in SCF data)
Identify f (θ) from empirical income distributions
→ Impute marginal tax using functional form (Gouveia/Strauss, 1994)
T (y )
−1/p
= b − b [sy p + 1]
y
(1)
→ Cagetti/DeNardi (2009) estimate b, s, p using PSID data
→ Back out marginal tax rates Ty0 and Tπ0 from (1),
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
18 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Calibration II
Constant elasticity disutility of effort ψ(e) = e 1+1/ε /(1 + 1/ε) with ε = .25
Cobb-Douglas technology Y (L, E ) = Lα E 1−α with α = .63 (workers’ share of
income in SCF data)
Identify f (θ) from empirical income distributions
→ Impute marginal tax using functional form (Gouveia/Strauss, 1994)
T (y )
−1/p
= b − b [sy p + 1]
y
(1)
→ Cagetti/DeNardi (2009) estimate b, s, p using PSID data
→ Back out marginal tax rates Ty0 and Tπ0 from (1),
→ w θ and w̃ θ from
y 1/ε
π 1/ε
1 − Ty0 (y ) =
and 1 − Tπ0 (π) =
,
wθ
w̃ θ
→ w and w̃ such that w̃ /w equals ratio of mean wages of entrepreneurs and
α
workers, and w̃ = (1 − α) (α/w ) 1−α
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
18 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Calibration III
Kernel estimate of inferred skill density f (θ), truncated at 99 percentile
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
19 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Calibration III
Kernel estimate of inferred skill density f (θ), truncated at 99 percentile
Iso-elastic cost distribution Gθ (φ) = (φ/φθ )η with η = .5
Adjust φθ to generate the pattern of the share of entrepreneurs in the right panel
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
19 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Redistribution Across Cost Types
Pareto weights G̃θ (φ) = Gθ (φ)ρΦ , ρΦ = 2
→ Redistribution from low to high cost agents (entrepreneurs to workers)
→ w falls by 10% as a result of tax policy
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
20 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Redistribution Across Skill Types
Pareto weights F̃ (θ) = F (θ)1/ρΘ , ρΘ = 2
→ Redistribution from high to low skill agents
→ w falls by 3% as a result of tax policy
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
21 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Redistribution Across θ and φ
Pareto weights ρΘ = 2, ρΦ = 2
→ Redistribution in both dimensions
→ w falls by 12% as a result of tax policy
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
22 / 29
Differential Tax Treatment of Profits and Income
Numerical Illustration: Higher ε
Pareto weights ρΘ = 2, ρΦ = 2, increased elasticity ε = .5 rather than ε = .25
→ Lower optimal marginal tax rates
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
23 / 29
Entrepreneurial Taxation with Credit Market Frictions
Introducing Credit Markets
Suppose each entrepreneur has to invest I to set up a firm
Entrepreneurs have no wealth, need to borrow funds from banks in competitive
credit market
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
24 / 29
Entrepreneurial Taxation with Credit Market Frictions
Introducing Credit Markets
Suppose each entrepreneur has to invest I to set up a firm
Entrepreneurs have no wealth, need to borrow funds from banks in competitive
credit market
→ Will the ‘right’ individuals become entrepreneurs in equilibrium?
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
24 / 29
Entrepreneurial Taxation with Credit Market Frictions
Introducing Credit Markets
Suppose each entrepreneur has to invest I to set up a firm
Entrepreneurs have no wealth, need to borrow funds from banks in competitive
credit market
→ Will the ‘right’ individuals become entrepreneurs in equilibrium?
→ Focus on occupational choice, fix effort
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
24 / 29
Entrepreneurial Taxation with Credit Market Frictions
Introducing Credit Markets
Suppose each entrepreneur has to invest I to set up a firm
Entrepreneurs have no wealth, need to borrow funds from banks in competitive
credit market
→ Will the ‘right’ individuals become entrepreneurs in equilibrium?
→ Focus on occupational choice, fix effort
• Workers supply fixed amount of labor l and get utility vW = wl
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
24 / 29
Entrepreneurial Taxation with Credit Market Frictions
Introducing Credit Markets
Suppose each entrepreneur has to invest I to set up a firm
Entrepreneurs have no wealth, need to borrow funds from banks in competitive
credit market
→ Will the ‘right’ individuals become entrepreneurs in equilibrium?
→ Focus on occupational choice, fix effort
• Workers supply fixed amount of labor l and get utility vW = wl
• Entrepreneurs hire labor and produce stochastic profits
π = Y (L) − wL + ,
∼ H (|θ)
H (|θ) MLRP H (|θ0 ) for θ > θ0
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
24 / 29
Entrepreneurial Taxation with Credit Market Frictions
Introducing Credit Markets
Suppose each entrepreneur has to invest I to set up a firm
Entrepreneurs have no wealth, need to borrow funds from banks in competitive
credit market
→ Will the ‘right’ individuals become entrepreneurs in equilibrium?
→ Focus on occupational choice, fix effort
• Workers supply fixed amount of labor l and get utility vW = wl
• Entrepreneurs hire labor and produce stochastic profits
π = Y (L) − wL + ,
∼ H (|θ)
H (|θ) MLRP H (|θ0 ) for θ > θ0
• Banks offer menus of credit contracts that supply funding I in return for
repayment schedule Rθ (π)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
24 / 29
Entrepreneurial Taxation with Credit Market Frictions
Introducing Credit Markets
Suppose each entrepreneur has to invest I to set up a firm
Entrepreneurs have no wealth, need to borrow funds from banks in competitive
credit market
→ Will the ‘right’ individuals become entrepreneurs in equilibrium?
→ Focus on occupational choice, fix effort
• Workers supply fixed amount of labor l and get utility vW = wl
• Entrepreneurs hire labor and produce stochastic profits
π = Y (L) − wL + ,
∼ H (|θ)
H (|θ) MLRP H (|θ0 ) for θ > θ0
• Banks offer menus of credit contracts that supply funding I in return for
repayment schedule Rθ (π)
R
⇒ Entrepreneurs’ expected utility (π − Rθ (π))dH(π|θ) − φ
⇒ Given any {Rθ (π)}, all entrepreneurs hire the same L s.t. YL = w
⇒ Can work with H(π|θ) directly, with support Π
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
24 / 29
Entrepreneurial Taxation with Credit Market Frictions
Credit Market Equilibrium I
Definition
A credit market equilibrium is a set of contracts {Rθ (π)} such that
(i)
Z
Z
(π − Rθ (π)) dH(π|θ) ≥
(π − Rθ0 (π)) dH(π|θ) ∀θ, θ0 ∈ Θ,
Π
Florian Scheuer (MIT)
(IC)
Π
Entrepreneurial Taxation
May 2010
25 / 29
Entrepreneurial Taxation with Credit Market Frictions
Credit Market Equilibrium I
Definition
A credit market equilibrium is a set of contracts {Rθ (π)} such that
(i)
Z
Z
(π − Rθ (π)) dH(π|θ) ≥
(π − Rθ0 (π)) dH(π|θ) ∀θ, θ0 ∈ Θ,
Π
(IC)
Π
(ii)
Z
Z
Rθ (π)dH(π|θ) − I dF (θ) ≥ 0
G (φ̃(θ))
Θ
(NNP)
Π
with
Z
(π − Rθ (π)) dH(π|θ) − vW ,
φ̃(θ) =
Π
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
25 / 29
Entrepreneurial Taxation with Credit Market Frictions
Credit Market Equilibrium I
Definition
A credit market equilibrium is a set of contracts {Rθ (π)} such that
(i)
Z
Z
(π − Rθ (π)) dH(π|θ) ≥
(π − Rθ0 (π)) dH(π|θ) ∀θ, θ0 ∈ Θ,
Π
(IC)
Π
(ii)
Z
Z
Rθ (π)dH(π|θ) − I dF (θ) ≥ 0
G (φ̃(θ))
Θ
(NNP)
Π
with
Z
(π − Rθ (π)) dH(π|θ) − vW ,
φ̃(θ) =
Π
(iii) there is no other set {R̃θ (π)} s.t., when offered in addition to {Rθ (π)}, earns
positive profits.
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
25 / 29
Entrepreneurial Taxation with Credit Market Frictions
Credit Market Equilibrium I
Definition
A credit market equilibrium is a set of contracts {Rθ (π)} such that
(i)
Z
Z
(π − Rθ (π)) dH(π|θ) ≥
(π − Rθ0 (π)) dH(π|θ) ∀θ, θ0 ∈ Θ,
Π
(IC)
Π
(ii)
Z
Z
Rθ (π)dH(π|θ) − I dF (θ) ≥ 0
G (φ̃(θ))
Θ
(NNP)
Π
with
Z
(π − Rθ (π)) dH(π|θ) − vW ,
φ̃(θ) =
Π
(iii) there is no other set {R̃θ (π)} s.t., when offered in addition to {Rθ (π)}, earns
positive profits.
Note: Allow for arbitrary sets of contracts, thus cross-subsidization not ruled out
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
25 / 29
Entrepreneurial Taxation with Credit Market Frictions
Credit Market Equilibrium I
Definition
A credit market equilibrium is a set of contracts {Rθ (π)} such that
(i)
Z
Z
(π − Rθ (π)) dH(π|θ) ≥
(π − Rθ0 (π)) dH(π|θ) ∀θ, θ0 ∈ Θ,
Π
(IC)
Π
(ii)
Z
Z
Rθ (π)dH(π|θ) − I dF (θ) ≥ 0
G (φ̃(θ))
Θ
(NNP)
Π
with
Z
(π − Rθ (π)) dH(π|θ) − vW ,
φ̃(θ) =
Π
(iii) there is no other set {R̃θ (π)} s.t., when offered in addition to {Rθ (π)}, earns
positive profits.
Note: Allow for arbitrary sets of contracts, thus cross-subsidization not ruled out
Restrict to contracts s.t. (i) 0 ≤ Rθ (π) ≤ π (limited liability), and
(ii) Rθ (π) non-decreasing (monotonicity)
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
25 / 29
Entrepreneurial Taxation with Credit Market Frictions
Credit Market Equilibrium II
Proposition
Under Assumption 1, the credit market equilibrium is such that only the single
contract R ∗ (π) = min{π, z ∗ } is offered and z ∗ solves
Z
Z
min{π, z ∗ }dH(π|θ) − I dF (θ) = 0
G (φ̃z ∗ (θ))
Θ
Π
with
Z
(π − min{π, z ∗ }) dH(π|θ) − vW ,
φ̃z ∗ (θ) =
Π
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
26 / 29
Entrepreneurial Taxation with Credit Market Frictions
Credit Market Equilibrium II
Proposition
Under Assumption 1, the credit market equilibrium is such that only the single
contract R ∗ (π) = min{π, z ∗ } is offered and z ∗ solves
Z
Z
min{π, z ∗ }dH(π|θ) − I dF (θ) = 0
G (φ̃z ∗ (θ))
Θ
Π
with
Z
(π − min{π, z ∗ }) dH(π|θ) − vW ,
φ̃z ∗ (θ) =
Π
The equilibrium is a pooling equilibrium with a standard debt contract offered
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
26 / 29
Entrepreneurial Taxation with Credit Market Frictions
Credit Market Equilibrium II
Proposition
Under Assumption 1, the credit market equilibrium is such that only the single
contract R ∗ (π) = min{π, z ∗ } is offered and z ∗ solves
Z
Z
min{π, z ∗ }dH(π|θ) − I dF (θ) = 0
G (φ̃z ∗ (θ))
Θ
Π
with
Z
(π − min{π, z ∗ }) dH(π|θ) − vW ,
φ̃z ∗ (θ) =
Π
The equilibrium is a pooling equilibrium with a standard debt contract offered
Intuition:
• By MLRP, low-skill borrowers have more probability weight in low-profit states
• Debt contracts put the maximal repayment in low-profit states
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
26 / 29
Entrepreneurial Taxation with Credit Market Frictions
Credit Market Equilibrium II
Proposition
Under Assumption 1, the credit market equilibrium is such that only the single
contract R ∗ (π) = min{π, z ∗ } is offered and z ∗ solves
Z
Z
min{π, z ∗ }dH(π|θ) − I dF (θ) = 0
G (φ̃z ∗ (θ))
Θ
Π
with
Z
(π − min{π, z ∗ }) dH(π|θ) − vW ,
φ̃z ∗ (θ) =
Π
The equilibrium is a pooling equilibrium with a standard debt contract offered
Intuition:
• By MLRP, low-skill borrowers have more probability weight in low-profit states
• Debt contracts put the maximal repayment in low-profit states
→ Debt contracts are least attractive to low-skill borrowers
→ Any deviation would attract a lower quality borrower pool and earn negative
profits
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
26 / 29
Entrepreneurial Taxation with Credit Market Frictions
Efficiency of Occupational Choice
Efficiency: type (θ, φ) should become entrepreneur if and only if
Z
πdH(π|θ) − I − φ ≥ vW
Π
⇒ Efficient critical cost value φ̃e (θ) =
Florian Scheuer (MIT)
R
Π
πdH(π|θ) − I − vW
Entrepreneurial Taxation
May 2010
27 / 29
Entrepreneurial Taxation with Credit Market Frictions
Efficiency of Occupational Choice
Efficiency: type (θ, φ) should become entrepreneur if and only if
Z
πdH(π|θ) − I − φ ≥ vW
Π
⇒ Efficient critical cost value φ̃e (θ) =
R
Π
πdH(π|θ) − I − vW
Corollary
There exists a skill-type θ̃ s.t.
R
Π
min{π, z ∗ }dH(π|θ̃) = I and
φ̃z ∗ (θ) > φ̃e (θ) ∀θ < θ̃
φ̃z ∗ (θ) < φ̃e (θ) ∀θ > θ̃
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
27 / 29
Entrepreneurial Taxation with Credit Market Frictions
Efficiency of Occupational Choice
Efficiency: type (θ, φ) should become entrepreneur if and only if
Z
πdH(π|θ) − I − φ ≥ vW
Π
⇒ Efficient critical cost value φ̃e (θ) =
R
Π
πdH(π|θ) − I − vW
Corollary
There exists a skill-type θ̃ s.t.
R
Π
min{π, z ∗ }dH(π|θ̃) = I and
φ̃z ∗ (θ) > φ̃e (θ) ∀θ < θ̃
φ̃z ∗ (θ) < φ̃e (θ) ∀θ > θ̃
Cross-subsidization in the credit market leads to occupational misallocation:
• Excessive entry of low-skilled types into entrepreneurship
• Insufficient entry of high-skilled types
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
27 / 29
Entrepreneurial Taxation with Credit Market Frictions
Efficiency of Occupational Choice
Efficiency: type (θ, φ) should become entrepreneur if and only if
Z
πdH(π|θ) − I − φ ≥ vW
Π
⇒ Efficient critical cost value φ̃e (θ) =
R
Π
πdH(π|θ) − I − vW
Corollary
There exists a skill-type θ̃ s.t.
R
Π
min{π, z ∗ }dH(π|θ̃) = I and
φ̃z ∗ (θ) > φ̃e (θ) ∀θ < θ̃
φ̃z ∗ (θ) < φ̃e (θ) ∀θ > θ̃
Cross-subsidization in the credit market leads to occupational misallocation:
• Excessive entry of low-skilled types into entrepreneurship
• Insufficient entry of high-skilled types
⇒ Too many and too few entrepreneurs simultaneously
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
27 / 29
Entrepreneurial Taxation with Credit Market Frictions
Corrective Tax Policy
Lemma
If the profit tax T (π) is such that after-tax profits π̂ ≡ π − T (π) are increasing, then
the credit market equilibrium given T (π) is a single debt contract RzT∗ (π̂) = min{π̂, zT∗ },
where zT∗ solves
Z
Z
G (φ̃zT∗ ,T (θ))
min{π − T (π), zT∗ }dH(π|θ) − I dF (θ) = 0,
Θ
and φ̃zT∗ ,T (θ) ≡
Π
R
Π
(π − T (π) − min{π − T (π), zT∗ }) dH(π|θ) − vW ∀θ ∈ Θ.
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
28 / 29
Entrepreneurial Taxation with Credit Market Frictions
Corrective Tax Policy
Lemma
If the profit tax T (π) is such that after-tax profits π̂ ≡ π − T (π) are increasing, then
the credit market equilibrium given T (π) is a single debt contract RzT∗ (π̂) = min{π̂, zT∗ },
where zT∗ solves
Z
Z
G (φ̃zT∗ ,T (θ))
min{π − T (π), zT∗ }dH(π|θ) − I dF (θ) = 0,
Θ
and φ̃zT∗ ,T (θ) ≡
Π
R
Π
(π − T (π) − min{π − T (π), zT∗ }) dH(π|θ) − vW ∀θ ∈ Θ.
Proposition
If the tax policy T (π) is introduced such that π − T (π) is increasing and, for all θ ∈ Θ,
Z
Z
T (π)dH(π|θ) = −
min{π − T (π), zT∗ }dH(π|θ) − I , then
Π
Π
(i) the resulting credit market equilibrium is s.t. φ̃zT∗ ,T (θ) = φ̃e (θ) for all θ ∈ Θ,
(ii) the gov’t budget is balanced.
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
28 / 29
Entrepreneurial Taxation with Credit Market Frictions
Corrective Tax Policy
Lemma
If the profit tax T (π) is such that after-tax profits π̂ ≡ π − T (π) are increasing, then
the credit market equilibrium given T (π) is a single debt contract RzT∗ (π̂) = min{π̂, zT∗ },
where zT∗ solves
Z
Z
G (φ̃zT∗ ,T (θ))
min{π − T (π), zT∗ }dH(π|θ) − I dF (θ) = 0,
Θ
and φ̃zT∗ ,T (θ) ≡
Π
R
Π
(π − T (π) − min{π − T (π), zT∗ }) dH(π|θ) − vW ∀θ ∈ Θ.
Proposition
If the tax policy T (π) is introduced such that π − T (π) is increasing and, for all θ ∈ Θ,
Z
Z
T (π)dH(π|θ) = −
min{π − T (π), zT∗ }dH(π|θ) − I , then
Π
Π
(i) the resulting credit market equilibrium is s.t. φ̃zT∗ ,T (θ) = φ̃e (θ) for all θ ∈ Θ,
(ii) the gov’t budget is balanced.
Regressive profit tax counteracts cross-subsidization and restores efficiency
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
28 / 29
Conclusion
Conclusion
Uniform profit and income taxation...
• ... provides some justification for trickle down based arguments
• ... calls for additional tax distortions, e.g. on inputs
Role of differential profit and income taxation in ...
• ... removing pecuniary externalities from uniform taxation
• ... correcting inefficient sorting into occupations with credit market frictions
Florian Scheuer (MIT)
Entrepreneurial Taxation
May 2010
29 / 29