Online Appendix for “A Road Map for Efficiently Taxing
Heterogeneous Agents”
Marios Karabarbounis
Federal Reserve Bank of Richmond
August 5, 2015
A Alternative models: Aggregate Effects
This section reports the aggregate effects of the optimal labor-income policy in the “Ex-
ogenous Human Capital” model and the “Constant Elasticity Model.”
Table 1 reports the effects and welfare gains for the model with exogenous human capital.
Overall, the aggregate effects are in the same range. In addition, the presence of endogenous
human capital seems to add 0.1% to our welfare gains. Table 2 compares the effects of the
the age- and assets- dependent tax reform for both models. Changes in macro variables and
welfare gains are close to the ones found in the benchmark economy. Table 2 also reports the
effects in macro aggregates and welfare gains for different values of labor supply elasticity.
1
A larger value of labor supply elasticity can increase labor supply (and capital) by a larger
amount. However, welfare gains seem robust across specifications.
1
In the benchmark model the (heterogeneity in) labor supply elasticity depends mostly on the distribution
of reservation wages and not on the value θ (see Hansen (1985), Rogerson (1988), and Chang and Kim (2006)).
In contrast, varying parameter θ
c
in the CEM will affect uniformly all agents.
1
Table 1: Aggregate Effects of Tax Reform: Benchmark and Exogenous Human Capital
Model Benchmark Exogenous Human Capital
K +19.7% +20.3%
L +2.7% +2.8%
C +6.4% +6.0%
w +5.6% +5.7%
r -1.0% -1.0%
Cons. Gini -0.2% -0.5%
CEV +1.5% +1.4%
Note: The Table reports the percentage change in aggregates due to the tax reform. The tax reform includes age, household
assets, and filing status as tax tags. Results are presented for the benchmark case and two alternative specifications. A model
with exogenous human capital accumulation and a model with constant elasticity of labor supply.
Table 2: Aggregate Effects of Tax Reform: Benchmark and CEM
Model Benchmark CEM
θ
c
= 0.4 θ
c
= 0.73 θ
c
= 1.5
K +17.7% +18.9% +18.4% +22.2%
L +0.6% +0.6% +0.9% +1.7%
C +3.7% +3.0% +3.1% +4.2%
w +5.8% +6.1% +5.9% +6.8%
r -1.1% -0.3% -0.3% -0.4%
Cons. Gini +0.3% +0.2% +0.2% +0.2%
CEV +1.0% +1.1% +1.1% +1.0%
Note: The Table reports the percentage change in aggregates due to the tax reform. The tax reform includes age and household
assets as tax tags. Results are presented for the benchmark case and a model with constant elasticity of labor supply. For the
latter case, I present the effects of the optimal tax system for a variety of labor supply elasticities: θ
c
= 0.4,θ
c
= 0.73, θ
c
= 1.5.
B Simpler Policies
In this section, I restrict the age-dependent tax system to simpler forms. In particular, I
analyze a linear form, τ
00
+ τ
01
j, a second-degree polynomial, τ
00
+ τ
01
j + τ
02
j
2
, and compare
them to our benchmark specification: τ
00
+ τ
01
j + τ
02
j
2
+ τ
03
j
3
. The aggregate effects and
2
optimal tax properties are shown in Table 3 and Figure 1, respectively.
Table 3: Age-dependent Taxes: Simpler Functional Forms
Linear Second-Degree Benchmark
Functional Form τ
00
+ τ
01
j τ
00
+ τ
01
j + τ
02
j
2
τ
00
+ τ
01
j + τ
02
j
2
+ τ
03
j
3
K +5.3% +5.0% +9.5%
L -1.0% -0.7% +0.8%
C +0.1% +0.4% +0.9%
w +2.2% +2.0% +4.1%
r -0.4% -0.4% +0.6%
Cons. Gini -1.1% -1.3% -1.7%
CEV +0.2% +0.3% +0.4%
Note: The table reports the percentage change in aggregates due to an age-dependent tax reform. Results are presented for a
linear tax function, a second-degree polynomial and the benchmark specification.
Figure 1: Optimal Age-dependent Taxes: Simpler Functional Forms
20 30 40 50 60 70
−0.08
−0.03
0.02
0.07
0.12
0.17
0.22
0.27
0.32
τ
0
Age
Optimal Tax by Age
Current U.S.
Linear
Second Degree
Benchmark
Note: The figure plots the properties of optimal age-dependent taxes for different functional forms: linear, second-degree
polynomial and benchmark case (third-degree polynomial).
A linear increasing function increases capital by a smaller amount compared to the bench-
mark. Moreover, labor supply decreases as the tax distortion induces older households to
retire earlier. If the tax function is a second-degree polynomial then the tax distortion also
increases but a smaller rate for older households. This makes the employment reduction small-
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er. Using the benchmark specification (third-degree polynomial), we can increase capital by a
larger amount but also increase employment. The flexibility allows to increase tax distortions
steeply up to age 50 and decrease taxes after that age. Although relatively small, welfare
gains double compared to a linear tax function.
C Household’s Problem: Value Functions
In this section, I write the value function for a household employing the female worker
V
{NE,E}
and a household whose members are not employed V
{NE,N E}
. The value function
for a household employing the female worker is:
V
{NE,E}
zj
(a, x, κ, E
−1
) = max
c,a
0
,h
f
(
log(c) + ψ
m
j
(1 − h
m
)
1−θ
1 − θ
+ ψ
f
j
(1 − h
f
)
1−θ
1 − θ
− ζ(E
f
−1
)
+βs
j+1
X
x
m
0
X
x
f
0
Γ
x
m
x
0
m
Γ
x
f
x
0
f
∗
[
(1 − λ
m
)
1 − p
X
s={2,3}
p
s
V
1
z(j+1)
(a
0
, x
0
, κ
s
, E) +
λ
m
1 − p
X
s={2,3}
p
s
V
{NE,N E}
z(j+1)
(a
0
, x
0
, κ
s
, E)]
(1)
s.t. h
f
= 0 (2)
(1+τ
c
)c +a
0
= (1−τ
ss
)W − T
L
(W ; S)+(1+r(1−τ
k
))(a +T r) (3)
E = {u, e} (4)
The value function for a household with no earners is:
4
V
{NE,N E}
zj
(a, x, κ, E
−1
) = max
c,a
0
(
log(c) + ψ
m
j
(1 − h
m
)
1−θ
1 − θ
+ ψ
f
j
(1 − h
f
)
1−θ
1 − θ
+βs
j+1
X
x
m
0
X
x
f
0
Γ
x
m
x
0
m
Γ
x
f
x
0
f
V
2
z(j+1)
(a
0
, x
0
, κ
0
, E)]
(5)
s.t. h
m
= 0, h
f
= 0 (6)
(1+τ
c
)c+a
0
= (1+r(1−τ
k
))(a+T r) (7)
E = {u, u} (8)
References
Chang, Y., and Kim, S.-B. (2006). “From individual to aggregate labor supply: a quantitative
analysis based on a heterogeneous agent macroeconomy”. International Economic Review,
47 (1), 1-27.
Hansen, G. D. (1985). “Indivisible labor and the business cycle”. Journal of Monetary
Economics, 16 (3), 309-327.
Rogerson, R. (1988). “Indivisible labor, lotteries and equilibrium”. Journal of Monetary
Economics, 21 (1), 3-16.
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