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Lecture 9: Optimal Taxation Extra slides
EC981 - Topics in Public Economics
Lecture 9: Optimal Taxation
Miguel Almunia
MSc in Economics - University of Warwick
Thursday, 6th March, 2014
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Lecture 9: Optimal Taxation Extra slides
Optimal Taxation
“Optimal” tax theory must combine lessons from excess burden
and tax incidence:
Optimal size of the pie? - Efficiency
How is the pie distributed? - Equity
What is the best way to design taxes given equity and
efficiency concerns?
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Lecture 9: Optimal Taxation Extra slides
Optimal Taxation
Efficiency perspective: finance the govt through lump-sum
taxation
Fixed amount per person regardless of individual characteristics
Does not induce behavioural responses, so no inefficiency
Equity perspective: individual lump-sum taxes
Tax higher-ability individuals a larger lump sum
Problem: we cannot observe individual abilities
Hence we tax outcomes, such as income or consumption
Creates distortions & inefficiency
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Lecture 9: Optimal Taxation Extra slides
Outline of Today’s Lecture
Traditional Optimal Taxation Theory:
Ramsey (1927): Inverse elasticity rule for taxation of goods
Mirrlees (1971): Optimal nonlinear income tax
Modern Optimal Taxation Theory:
Saez (2001): optimal tax rate for top earners
Borrows some results from Diamond (1998)
* Diamond & Saez (2011): survey of the literature
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Lecture 9: Optimal Taxation Extra slides
Traditional Approaches
Ramsey (1927): linear tax systems
T (z) = t · z
Rules out lump-sum taxes by assumption
Homogeneous agents
Mirrlees (1971): nonlinear tax systems
T (z) = f (z)
Permits lump sum taxes
Heterogeneity in agents’ skills
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Tax Problem
Government levies taxes on goods in order to accomplish two
goals:
1 Raise total revenue (R) of amount E
2 Minimize utility loss for agents in the economy
Representative individual
No redistributive concerns
As in efficiency analysis, assume that individual does not
internalize the effect of τi on govt budget
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Tax Problem
United Kingdom, 1903-1930
Mathematician & Philosopher
Cambridge University
.
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Assumptions
1 No lump-sum or nonlinear taxes - Only proportional taxes
2 Cannot tax all commodities (eg, leisure untaxed)
3 Production prices fixed (normalized to one):
pi = 1
⇒ qi = 1 + τi
Note: full mathematical derivation in “extra slides” at the end
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Graphical IntuitionTaxation of elastic vs. inelastic goods
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Interpretation & Limitations
Intuition: tax inelastic goods to minimize efficiency costs
Limitations:
Does not take into account redistributive motives
Necessities usually more inelastic than luxuries
Thus, optimal Ramsey tax system is regressive
Restricted to linear tax systems
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Lecture 9: Optimal Taxation Extra slides
Application of Ramsey to Taxation of Savings
Standard lifecycle model:
max∑
t
ut (ct) subj. to qtct = W
where qt = (1 + τt) pt
Consumption in each period treated like different commodities
Let capital income tax be θ, levied on interest rate:
qt =1
(1 + (1 − θ) r)t
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Lecture 9: Optimal Taxation Extra slides
Application of Ramsey to Taxation of Savings
For any θ > 0, implied optimal tax goes to infinity:
qt
pt= 1 + τt =
(1 + r
1 + (1 − θ) r
)t
⇒ limt→∞
τt = ∞
But Ramsey formula implies that optimal tax τ∗ cannot be ∞
for any good
Therefore, optimal capital income tax must be zero in long run
(Judd 1985; Chamley 1986)
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Lecture 9: Optimal Taxation Extra slides
Zero Capital Taxation Result: Limitations
The zero result within Ramsey model no longer holds when
assumptions are relaxed, for example:
Allowing for progressive income taxation
Allowing for credit market imperfections
Finitely-lived agents with finite bequest elasticities
Allowing for myopic agents
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Lecture 9: Optimal Taxation Extra slides
Mirrlees (1971): Incorporating Behavioural Responses
Ramsey’s approach ignored equity-efficiency tradeoff
Mirrlees (1971) incorporates behavioural responses:
1 Standard labour supply model2 Individuals differ in ability w
3 Govt maximizes social welfare function (SWF) subj. to:
Budget constraint
Behavioural responses to taxation
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Lecture 9: Optimal Taxation Extra slides
Mirrlees (1971): Incorporating Behavioural Responses
Scotland, 1936
Professor of Economics
Cambridge University
1996 Nobel Prize Winner
.
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Lecture 9: Optimal Taxation Extra slides
Mirrlees (1971): Model
Individual maximizes:
u (c , l) subj. to c = wl − T (wl)
Ability distribution given by density f (w)
Government maximizes:
SWF =
ˆ
G [u (c , l)] f (w) dw
subj. to budget
ˆ
T (wl) f (w) dw ≥ E
subj. to indiv FOC w(
1 − T ′ (·))
uc − ul = 0
where G (·) is increasing and concave
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Lecture 9: Optimal Taxation Extra slides
Mirrlees (1971): Model
Govt maximizes weighted sum of utilities of ex-post
consumption
With equal weights and diminishing marginal utility, we would
equalize everyone’s income
Utilitarianism leads to comunism!
Is maximizing total ex-post utility the right objective function?
Deep debate dating back to Rawls, Nozick, Sen...
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Lecture 9: Optimal Taxation Extra slides
Mirrlees (1971): Results
Mirrlees formulas are complicated, only a few general results:
1 T ′ (·) ≤ 1: Obvious, because otherwise no one works2 T ′ (·) ≥ 0: Non-trivial. Rules out EITC (incentives for labour
force participation with tax subsidies)3 T ′ (·) = 0 at the bottom of the skill distribution (assuming
everyone works)4 T ′ (·) = 0 at the top of the skill distribution (if skill
distribution is bounded)
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Lecture 9: Optimal Taxation Extra slides
Mirrlees (1971): Results
Mirrlees model had huge impact in fields like contract theory
Models with asymmetric information
But little impact on practical tax policy
Recently, connected to empirical tax literature:
Diamond (AER, 1998), Saez (REStud, 2001)
Sufficient statistic formulas in terms of elasticities
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Lecture 9: Optimal Taxation Extra slides
Optimal Income Taxation: Sufficient Statistic Formulas
Revenue-maximizing linear tax: Laffer curve
Top income tax rate (Saez 2001)
Overview of this literature: *Diamond and Saez (JEL, 2011)
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Lecture 9: Optimal Taxation Extra slides
Laffer Curve: Revenue Maximizing Rate
Useful benchmark for optimal rate
Let tax revenue be R (τ) = τ · z (t − τ)
Notice: reported income z is a function of net-of-tax rate
(1 − τ)
R (τ) has an inverse U-shape:
No taxes: R (τ = 0) = 0
Confiscatory taxes: R (τ = 1) = 0
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Lecture 9: Optimal Taxation Extra slides
Laffer Curve: Revenue Maximizing Rate
Revenue maximizing rate, τ∗:
R ′ (τ∗) = 0
z − τdz
d (1 − τ)= 0
z
[(1 − τ)
z
]
− τdz
d (1 − τ)
[(1 − τ)
z
]
︸ ︷︷ ︸
ε
= 0
1 − τ − τε = 0
⇒ τ∗ =1
1 + ε
Strictly inefficient to have τ > τ∗ (Why?)
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Lecture 9: Optimal Taxation Extra slides
Laffer Curve: Revenue Maximizing RateGraphical representation
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Lecture 9: Optimal Taxation Extra slides
Saez (2001): Using Elasticities to Derive Optimal Tax Rates
Derive optimal tax rate τ using perturbation argument
Assume there are no income effects: εC = εU = ε
Diamond (1998) shows this is a key theoretical simplification
Assume that there are N individuals above earnings z
Let zm (1 − τ) be avge income function of these individuals
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Lecture 9: Optimal Taxation Extra slides
Saez (2001): Optimal Income Tax Rate
Three effects of small ∆τ > 0 reform above z :
1 Mechanical increase in tax revenue:
∆M = N · [zm − z ]∆τ
2 Behavioural response:
∆B = Nτ∆zm = Nτ
(
−∆τ∆zm
∆ (1 − τ)
)
= −Nτ
1 − τεzm∆τ
3 Welfare effect: if govt values rich consumption at g ∈ (0, 1):
∆W = −gdM
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Lecture 9: Optimal Taxation Extra slides
Saez (2001): Optimal Income Tax Rate
$%&'(&)*+,!-./(0,/1234526
78,39):!%./(0,!22;
2;3452;6
<
=,/>).%/)+!9):!%./8,)&,?@232;A
B,>)C%(8)+!8,&'(.&,!9):!+(&&?!2 1!3 ,!2! D5E3 6
2
4('!*8)/F,9?!&+(',!E3 )*(C,!2;!!G,H(80?!&+(',!E3 )*(C,!2;!
Source: Diamond and Saez (JEL, 2011)
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Lecture 9: Optimal Taxation Extra slides
Saez (2001): Optimal Income Tax Rate
Optimal tax rate solves:
∆M +∆W +∆B = 0
After some algebra:
τ∗top
1 − τ∗top=
(1 − g) [(zm/z)− 1]
εzm/z
Top tax rate τ∗top is higher when:
↓ g : less weight on welfare of the rich
↓ ε: lower elasticity of taxable income
↑ zm/z : higher income inequality
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Lecture 9: Optimal Taxation Extra slides
Saez (2001): Optimal Income Tax Rate
With Pareto distribution, zm/z converges to(
aa−1
)
, where a
is Pareto distn. parameter
Simplified formula is then:
τ∗top =1 − g
1 − g + εa
In the US, zm/z ≈ 3 and quite stable over time
So Pareto distn. parameter is zm
z= 3 =
aa−1
⇒ a = 1.5
We can estimate ε
Society decides value of g (relative weight of rich on SWF)
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Lecture 9: Optimal Taxation Extra slides
US Income Distribution approximated by Pareto Distribution
11.
52
2.5
Em
piric
al P
aret
o C
oeffi
cien
t
0 200000 400000 600000 800000 1000000z* = Adjusted Gross Income (current 2005 $)
a=zm/(zm-z*) with zm=E(z|z>z*) alpha=z*h(z*)/(1-H(z*))
Source: Diamond and Saez (JEL, 2011)
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Lecture 9: Optimal Taxation Extra slides
Zero Top Rate with Bounded Skill Distribution
Suppose top earner make zT , second makes zS
Tax only raised on top earner, so zm = zT :
∆M = N∆τ [zm − z ] → 0 < ∆B = N∆τ · ε ·τ
1 − τzm
Optimal τ = 0 for top earner
Standard Mirrleesian result
But the result applies only to the top earner
No practical relevance
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Lecture 9: Optimal Taxation Extra slides
Connection to Revenue Maximizing Tax Rate
Revenue-maximizing top tax rate can be calculated by setting
g = 0
Utilitarian SWF: g = uc (zm) → 0 when z → ∞
Rawlsian SWF: g = 0 for any z > min (z)
If g = 0, we obtain τ∗top = τmax = 11+εa
Assuming a = 1.5:
ε = 0.2 ε = 0.5 ε = 1
g = 0 0.77 0.57 0.40
g = 0.5 0.62 0.40 0.25
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Lecture 9: Optimal Taxation Extra slides
Mathematical Derivation of Ramsey (1927)
Note: the following slides draw heavily on Raj Chetty’s lecture
slides for his Public Economics course at Harvard University,
available at www.rajchetty.com
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Individual’s problem
Individual maximizes utility over goods (leisure untaxed):
max{x1,..xN ,l}
u (x1, ..., xN , l)
subj. to q1x1 + ...+ qNxN = wl + Y
Lagrangian:
L = u (x1, ..., xN , l) + α (wl + Y − q1x1 − ...− qNxN)
Solve and obtain indirect utility V (q,Y )
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Government’s problem
Then government maximizes welfare of representative
individual:
max V (q,Y )
subject to the tax revenue requirement:
N∑
i=1
τixi (q,Y ) ≥ E
Government’s Lagrangian:
L = V (q,Y ) + λ
[
E −
N∑
i=1
τixi (q,Y )
]
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Government’s problem
L = V (q,Y ) + λ
[
E −
N∑
i=1
τixi (q,Y )
]
⇒∂L
∂qi=
∂V
∂qi︸︷︷︸
Loss of indiv. welfare
+λ
⎡
⎢⎢⎢⎢⎢⎢⎣
xi︸︷︷︸
Mech. Effect
+∑
j
τj∂xj
∂qi
︸ ︷︷ ︸
Behav. Response
⎤
⎥⎥⎥⎥⎥⎥⎦
=
Using Roy’s Identity(∂V∂qi
= −αxi
)
:
(λ− α) xi + λ∑
j
τj∂xj
∂qi= 0
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Perturbation ArgumentEquivalent but more intuitive than original Ramsey analysis
Effect of tax increase (τi to τi + dτi) on social welfare:
Marginal effect on govt revenue:
dR = xidτi︸ ︷︷ ︸
mech. effect
+∑
j
τjdxj
︸ ︷︷ ︸
behav. response
Marginal effect on private surplus:
dU =∂V
∂qidτi = −αxidτi
Optimum characterized by balancing the two marginal effects:
dU + λdR = 0
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Equilibrium
After some algebra, arrive at the following expression for the
optimal tax rates, τ∗j :
∑
j
τj∂xj
∂qi= −
xi
λ(λ− α)
N equations in N unknowns
Notice the relevance of cross-price elastiticies
λ = Lagrange multiplier in the govt’s problem
α = Lagrange multiplier in the individual’s problem (mgl utility
of money)
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Compensated Elasticities
Starting from the perturbation argument formula:
dU + λdR = 0
Rewrite in terms of Hicksian elasticities and use Slutsky
equation:∂xj
∂qi=
∂hj
∂qi− xi
∂xj
∂Y
Combining the two equations:
(λ− α) xi + λ∑
j
τj
(∂hj
∂qi− xi
∂xj
∂Y
)
= 0
⇒1
xi
∑
j
τj∂hj
∂qi= −
θ
λ
where θ = λ− α− λ∂!
j τjxj
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Compensated Elasticities
θ is independent of i and measures the value for the govt of
introducing a $1 lump sum tax:
θ = λ− α− λ∂∑
j τjxj
∂Y
Three effects of introducing a $1 lump sum tax:
1 Direct value of λ for the govt2 Loss in welfare of α for the individual3 Behavioural effect → Loss in tax revenue:
∂!
j τjxj
∂Y
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Index of Discouragement
1
xi
∑
j
τj∂hj
∂qi=
θ
λ
Suppose revenue requirement E is small, so all taxes are small
Then tax τj on good j reduces consumption of good i (holding
utility constant) by approx
dhi = τj∂hi
∂qj
Numerator of LHS (top equation): total reduction in cons of
good i
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Index of Discouragement (cont’d)
Dividing by xi yields % reduction in consumption of each good
i – “index of discouragment” of the tax system on good i
Ramsey tax formula says the indices of discouragement must
be equal across goods at the optimum
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Lecture 9: Optimal Taxation Extra slides
Ramsey (1927): Inverse Elasticity Rule
Introducing elasticities, we can write Ramsey formula as:
N∑
j=1
τj1 + τj
εcij =
θ
λ
θ = λ− α− λ∂!
j τj xj
∂Yis the value for the govt of introducing
a $1 lump sum tax
Consider special case where εcij = 0 for all i = j
Slutsky matrix is diagonal
Obtain classic inverse elasticity rule:
τi1 + τi
=1
εcii
θ
λ