Optimal Taxation in Life-Cycle Economies
Journal of Economic Theory 105, 338-369 (2002) Andres Erosa and Martin Gervais Presented by Chung Tran Macro Group Meeting - 10 Sept 2008
Erosa & Gervais ()
Optimal Taxation
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Classic Questions in Public Finance
How to nance a given streams of government spending in the absence of lump-sum taxation? Which goods should be taxed: capital/labor/consumption?
Erosa & Gervais ()
Optimal Taxation
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Chamley-Judd's Classic Results: Negative
Neoclassical growth model: Households: representative in nitely-lived horizon agents Firms: Cobb-Douglass production technology Government: exogenous streams of government purchases and there is no lump-sump taxation Deterministic model with complete markets
Main results Con scatory capital income tax in the short run Zero capital income tax in the long-run
Erosa & Gervais ()
Optimal Taxation
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Is Nonzero Capital Income Tax Optimal? Incomplete market and dynamic ine ciency: Idiosyncratic shocks and borrowing constraint (Hubbard and Judd (1986), Aiyagari (1995), Imrohoroglu(1998)) Human capital accumulation and borrowing constraint (Boldrin and Montes (2002))
Life-cycle structure Alvarez et al. (1992) Erosa and Gervais (2002) Garriga (2003)
Hybrid quantitative models Fuster, Imrohoroglu and Imrohoroglu (2007) Conesa, Kitao and Kruger (2008) Optimal capital income tax rate is signi cantly positive at 36%. Optimal labor income tax is a at tax of 23% with $7200 deductible.
Erosa & Gervais ()
Optimal Taxation
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Erosa and Gervais's Main Findings
Tax consumption and labor income at dierent rates over an individual's lifetime If impossible, nonzero tax on capital income is also optimal.
Erosa & Gervais ()
Optimal Taxation
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Mechanism Corlett-Hague's intuition: the degree of substitutability between taxed and untaxed goods and tax rate Individual's e ciency units are age-dependent: hump-shape. Optimal consumption - work plan is almost never constant over individual's life-cycle. The degree of substitutability between consumption and leisure changes as individuals age. Optimal tax rule: tax rates on consumption and labor income should be conditioned on age. Tax capital income is an alternative way to mimic that rule.
Erosa & Gervais ()
Optimal Taxation
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Overlapping Generation Economy Economy as a Spreadsheet
Calender Time Generations t-J t-J-1 t-J-2 t-J-3
t=0 t=1 t=2 ..
t-1 t
1 0
# of Gen
J J-1 J-2 J-3 J-4
t=J+1
…
J …
J
J J-1 J J-2 J-1 J J-3 J-2 J-1 J
2 1 0
3 2 1 0
4 3 2 ..
5 4 3 ….
6 5 4
7 6 5
8 7 6
9 8 7
10 9 8
… 10 9
J … 10
J+1 J+1
Note: t and j are calendar time and age
Erosa & Gervais ()
Optimal Taxation
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Households
max
J X j =j0 (t )
β j −j0 (t ) U
ct ,j (t ) , ..., 1 − lt ,J ,
subject to the a sequences of budget constraint
ct , j + a t , j + 1
=
wt ,j zj lt ,j + (1 + rt ,j ) at ,j ,
where, wt ,j : age-dependent wage rate rt ,j : age dependent interest rate j = j0 (t ) , ..., J ,and aj ,j0 (t ) = 0 if t ≥ 0.
Erosa & Gervais ()
Optimal Taxation
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Technology and Feasibility
Production function
yt = f (kt , lt ) .
Aggregate resource constraint
ct + (1 + n) kt +1 − (1 − δ) kt + gt ≤ y , where ct =
J X
µj ct −j ,j and
lt =
j =0
Erosa & Gervais ()
J X
µj lt −j ,j .
j =0
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Government Government budget constraint (1 + b rt ) bt + gt = (1 + n) bt +1 + τtk−1,j µj at −j ,j +
J X j =0
τtw−1,j µj zj lt −j ,j .
The government's objective function ∞ X
γ t Ut ,
t =−J
where γ : inter-generational discount factor 0 < γ < 1 and Ut : the indirect utility function of generation t .
Erosa & Gervais ()
Optimal Taxation
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Factor prices Market factor prices
rbt wct
= =
fk (kt , lt ) − δ, fl (kt , lt ) .
Individual's age-dependent factor prices (after paying tax)
wt − j ,j rt −j ,j
Erosa & Gervais ()
= =
1 − τtw−j ,j wbt ,
1 − τtk−j ,j brt .
Optimal Taxation
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Households' Optimal Allocation Taking the FOCs yields the following optimal conditions
Uct,j − pt ,j Ult,j + pt ,j wt ,j zj −pt ,j + pt ,j +1 (1 + rt ,j +1 ) at , J + 1
= 0, ≤ 0, = 0, = 0,
where pt ,j is Lagrangian multiplier. Assuming nonzero labor supply one could have the solution to the household problem as
Ult,j Uct,j Uct,j Uct,j +1 Erosa & Gervais ()
=
wt ,j zj ,
= (1 + rt ,j +1 ) . Optimal Taxation
(1) (2) Presented by Chung Tran Macro Group / 22
Ramsey Problem: Primal Approach The Ramsey problem in this life-cycle economy is given by ∞ X
max J
γ o∞ n {ct ,j ,lt ,j }j =j0 (t ) ,kt +J +1 t =−J t =−J
t
Wt .
subjected to the implemetability constraint associated with the cohort born in period t J X j =j0 (t )
Uct,j ct ,j + Ult,j lt ,j
= Uct ,j0 (t ) 1 + rt ,j0 (t )
at ,j0 (t ) = At ,j0 (t ) ,
where, the pseudo-welfare function Wt is de ned as
Wt =
J X
β j −j0 (t )
j =j0 (t )
Erosa & Gervais ()
U ct ,j (t ) , ..., 1 − lt ,J ,
Optimal Taxation
+ λt
Uct,j ct ,j + Ult,j lt ,j
− λt At ,j0
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The Ramsey Allocation Assuming positive labor supply one could obtain the following optimal condition for the consumption-work plan −
Wlt,j Wct,j
=
(1 + λt ) Ult ,j + λt Ult ,j Htl ,j
(1 + λt ) Uct ,j + λt Uct ,j Htc,j
b t +j , = zj w
(3)
for sequence of consumption
Wct,j Wct,j +1
=
(1 + λt ) Uct ,j + λt Uct ,j Htc,j
(1 + λt ) Uct ,j +1 + λt Uct ,j +1 Htc,j +1
= 1 +b rt + j + 1 .
(4)
where,
H
c t ,j
H
l t ,j
Erosa & Gervais ()
PJ =
i =j0 (t )
PJ =
i =j0 (t )
Uct,i , ct,i ct ,i + Ult,i ,ct,i lt ,i , Uct,j Uct,i , lt,i ct ,i + Ult,i ,lt,i lt ,i . Ult,j
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Labor Income Tax Rate Combining (1) and (3) yields τtw,j =
Htl ,j − Htc,j . 1 + λt + λt Htl ,j
λt
(5)
Labor income tax rates are given by
Erosa & Gervais ()
Htl ,j = Htc,j , 0 if Htl ,j > Htc,j , 0 if Htl ,j < Htc,j .
τtw,j
= 0 only if
τtw,j
>
τtw,j
<
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Capital Income Tax Rate Combining (2) and (4) yields 1 + λt + λt Htc,j 1 + brt +j +1 = . c 1 + 1 − τtk,j +1 brt +j +1 1 + λt + λt Ht ,j +1
(6)
Optimal capital income tax rates are given by
Erosa & Gervais ()
Htc,j = Htc,j +1 , 0 if Htc,j > Htc,j , 0 if Htc,j < Htc,j .
τtk,j
= 0 if
τtk,j
>
τtk,j
<
Optimal Taxation
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Dierent Forms of Utility Function CES Utility h
U (ct ,j , lt ,j ) =
c
ρ t ,j
+θ 1−l
Additively Separable Utility
ρ t ,j
i 1−σ ρ
1−σ
U (ct ,j , lt ,j ) = u (ct ,j ) + v (lt ,j ) Cobb-Douglass Utility h
U (ct ,j , lt ,j ) = u (ct ,j ) v (lt ,j ) =
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Optimal Taxation
ct ,j (1 − lt ,j )θ
i1−σ
1−σ Presented by Chung Tran Macro Group / 22
Steady State Tax Rates with Cobb-Douglass Utility Labor income tax rates τjw
1−
τjw
=
λ 1 + λ − lj [1 + λ (1 − σ) (1 + θ)] − λσ
Age-dependent The more labor supply the higher rate
Capital income tax rates 1−σ) 1 + λ + λ −σ−θ( 1 + br 1−lj = −σ−θ(1−σ) 1 + 1 − τjk br 1 + λ + λ 1−lj +1
Age-dependent b τ > 0 if 1+ 11+ > 1 or ( −τ )b k
j
Erosa & Gervais ()
r
k j
r
lj +
1
> lj
Optimal Taxation
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CES Utility
FIG. 1.
Labor supply and tax rates over the lifetime of individuals.
Erosa & Gervais ()
Optimal Taxation
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Additive Separable Utility
FIG. 2.
Labor supply, productivity, and tax rate over the lifetime of individuals.
Erosa & Gervais ()
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Cobb-Douglass Utility
FIG. 3.
Labor supply and tax rates over the lifetime of individuals.
Erosa & Gervais ()
Optimal Taxation
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Conclusion
Capital Income Tax: NOT too bad in life-cycle economies! It imitates tax on consumption/leisure that tend to increase as individual ages. It imitates labor income labor income that declines with age.
Erosa & Gervais ()
Optimal Taxation
Presented by Chung Tran Macro Group / 22
