Optimal Taxation
Reference: L&S 3rd edition chapter 16
1
Warm-Up: The Neoclassical Growth Model with Endogenous Labour Supply
• You looked a little bit at this for Problem Set 3.
• Study planner’s problem:
∞
X
max
{ct ,Lt ,it ,kt+1 }∞
t=0
β t u (ct , Lt )
t=0
s.t.
Kt+1 = F (Kt , Lt ) + (1 − δ) Kt − ct
K0 given
• FOC directly from the sequence problem.
• ct :
β t uc (ct , Lt ) − λt = 0
(1)
−λt + λt+1 [FK (Kt , Lt ) + (1 − δ)] = 0
(2)
β t uL (ct , Lt ) + λt FL (Kt , Lt ) = 0
(3)
• Kt+1 :
• Lt :
• From (1) and (3) derive an intratemporal condition:
FL (Kt , Lt ) = −
1
uL (ct , Lt )
uc (ct , Lt )
Econ 210, Fall 2013
Pablo Kurlat
– In a competitive equilibrium, this will be equal to the wage
• From (1) and (2) derive the Euler equation:
uc (ct , Lt )
= [FK (Kt+1 , Lt+1 ) + (1 − δ)]
βuc (ct+1 , Lt+1 )
– In a competitive equilibrium this will be the interest rate
• From the time-zero perspective:
t−1
β t uc (ct , Lt ) Y
1
=
≡ pt
uc (c0 , L0 )
F (Ks+1 , Ls+1 ) + (1 − δ)
s=0 K
2
Setup of the Optimal Taxation Problem
• Technology and preferences as in the Neoclassical Growth Model
F (Kt , Lt )
X
β t u (ct , Lt )
t
• Aggregate resource constraint
ct + gt + K t+1 ≤ F (Kt , Lt ) + (1 − δ)Kt
• gt exogenous
• Linear taxes on labour income
τ lt
• Linear taxes on capital income
τ kt
• (Taxes on consumption are redundant)
• We do not allow a lump-sum tax
• The government has an initial debt equal to B0 plus interest, i.e. R0 B0
• The price of consumption in period t is
pt ≡

Qt−1
if t ≥ 1
1
if t = 0
1
s=0 Rs+1
2
(4)
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• Competitive markets with wages wt and rental rate rK
t
• How to find the optimal taxes?
2.1
Households, government and firms
• Government budget constraint
Bt+1 = g t − τ l t wt Lt − τ k t (rtk − δ)Kt + Rt Bt
• You will sometimes see this written as τtk rtK instead of τtk rtK − δ , i.e. the government taxes
the gross rather than the net return on capital. It doesn’t make much difference.
• Together with a no-Ponzi condition this reduces to
X
pt g t − τ l t wt Lt − τ k t (rtk − δ)Kt ≤ −R0 B0
t
• Household budget constraint
At+1 = wt 1 − τ l t Lt − ct + Rt At
• Together with a no-Ponzi condition this reduces to
X
pt ct − wt 1 − τ l t Lt ≤ R0 A0
t
• Arbitrage between bonds and capital implies
Rt = 1 + (1 − τ k t )(rtk − δ)
(5)
i.e. the interest rate equals the after-tax return on capital
– Depending on how we set things up, we might not want to impose this arbitrage condition in period 0.
• Capital-income taxes are implemented this way:
– The goverment taxes the net (i.e. after depreciation) rental of capital
– If household A lends to household B, there is no tax on that
– Also, there is no tax on interest on government bonds
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– (This is without loss of generality - why?)
– But arbitrage between capital and bonds implies that equilibrium interest rates are
equated to after-tax returns on capital (5)
• Market clearing:
At = Bt + Kt
• Firm profits:
π(Kt , Lt ) = F (Kt , Lt ) − wt Lt − rtk Kt
2.2
Equilibrium conditions
Definition 1. A competitive equilibrium is a policy g t , τ k t , τ l t , an allocation {ct , K t+1 , Lt } and
prices wt , rK
t , pt , such that households maximize utility s.t. budget constraint, firms maximize
profits, the government budget constraint holds and markets clear.
• Each policy (as long as it balances the budget) defines a different equilibrium.
• Some policies will be optimal (acoording to some optimality criterion), some will not.
• In principle, to find optimal policies we need to find allocations for every possible τ k t , τ l t and
then optimize over τ k t , τ l t .
• Instead, work with allocations directly. The “Primal Approach” or “Ramsey Approach”
3
The Primal Approach
• For any given policy, we can describe the equilibrium conditions:
• Firm FOC:
rtk = FK (Kt , Lt )
(6)
wt = FL (Kt , Lt )
(7)
• Consumer FOC
β t uc (ct , Lt ) − λpt = 0
β t uL (ct , Lt ) + λpt 1 − τ l t wt = 0
• From these, derive two conditions
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• Intratemporal:
β t uL (ct , Lt )
β t uc (ct , Lt )
=−
pt
pt (1 − τ l t ) wt
uL (ct , Lt )
uc (ct , Lt ) = −
(1 − τ l t ) wt
uL (ct , Lt )
wt 1 − τ l t = −
uc (ct , Lt )
(8)
• Intertemporal:
β t uc (ct , Lt )
= uc (c0 , L0 )
pt
β t uc (ct , Lt )
pt =
uc (c0 , L0 )
(9)
• Back to household budget
X
pt ct − wt 1 − τ l t Lt = R0 (B0 + K0 )
t
We write it with equality because in equilibrium it must hold with equality (households won’t
leave money on the table)
• Replace prices with the household FOC:
X β t uc (ct , Lt ) uc (c0 , L0 )
t
X
uL (ct , Lt )
ct +
Lt = R0 (B0 + K0 )
uc (ct , Lt )
(10)
β t [uc (ct , Lt ) ct + uL (ct , Lt ) Lt ] = uc (c0 , L0 ) R0 (B0 + K0 )
t
• No prices! No taxes! (Except for initial date). (10) known as an implementabilty condition
Proposition 1. An allocation {ct , K t , Lt } can be part of a competitive equilibrium iff (4) and
(10) hold with equality.
Proof.
• Only if: shown above
• If: Start with an allocation that satisfies (4) and (10)
1. Construct prices and taxes
(a) Find rkt and wt from (6) and (7)
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(b) Find pt from (9). (This also tells you Rt )
(c) Find τ l t from (8)
(d) Find τ k t from (5)
2. Check for equilibrium
(a) Factor prices imply firm optimization and zero profits
(b) Prices and taxes imply household FOCs hold
(c) (10) implies that household budget constraint holds with equality ⇒ household
optimization
(d) Use the budget constraint, constant returns to scale and the resource constraint:
X
pt ct − wt 1 − τ l t Lt = R0 (B0 + K0 )
t
X
pt ct − FL (Kt , Lt ) Lt + τ l t FL (Kt , Lt ) Lt = R0 (B0 + K0 )
(use factor prices)
pt ct − Yt + FK (Kt , Lt ) Kt + τ l t FL (Kt , Lt ) Lt = R0 (B0 + K0 )
(constant returns)
⇒
t
⇒
X
t
"
⇒
X
t
"
pt
−g t − Kt+1 + (1 − δ) Kt +
FK (Kt , Lt ) Kt + τ l t FL (Kt , Lt ) Lt
#
= R0 (B0 + K0 )
#
(FK (Kt , Lt ) − δ) Kt + τ l t FL (Kt , Lt ) Lt − g t +
⇒
pt
= R0 (B0 + K0 )
k
1
−
τ
(F
(K
,
L
)
−
δ)
+
1
K
−
K
K
t
t
t
t+1
t
" t #
k
P
pt τt (FK (Kt , Lt ) − δ) Kt + τ l t FL (Kt , Lt ) Lt − g t +
t
k
= R0 (B0 + K0 )
Kt+1 −pt + 1 − τt+1
(FK (Kt+1 , Lt+1 ) − δ) + 1 pt+1
k
+ 1 − τ0 (FK (K0 , L0 ) − δ) + 1 K0
" #
X pt τtk (FK (Kt , Lt ) − δ) Kt + τ l t FL (Kt , Lt ) Lt − g t
= R0 B0
⇒
+Kt+1 [−pt + Rt+1 pt+1 ]
t
X
τtk
(resource constraint)
(rearranging)
(rearranging)
(use Rt = 1 + 1 − τtk (FK (Kt , Lt ) − δ) )
X pt
)
⇒
pt τtk (FK (Kt , Lt ) − δ) Kt + τ l t FL (Kt , Lt ) Lt − g t = R0 B0 (use pt+1 =
R
t+1
t
so the government budget constraint holds.
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Optimal Taxes
• The government’s problem is
X
max
ct ,Lt ,K t+1 ,τ0k
β t u (ct , Lt )
t
s.t. ct + gt + K t+1 = F (Kt , Lt ) + (1 − δ)Kt
X
β t [uc (ct , Lt ) ct + uL (ct , Lt ) Lt ] = uc (c0 , L0 ) R0 [B0 + K0 ]
t
• Suppose τ0k is fixed for now.1 Let µ be the multiplier on the implementability constraint.
Note: the set of allocations that satisfy implementability constraint may not be convex.
• Define
W (c, L) ≡ u(c, L) + µ [uc (c, L) c + uL (c, L) L]
• Problem becomes
max
ct ,K t+1 ,τ0k
X
β t W (ct , Lt ) − µuc (c0 , L0 ) R0 [B0 + K0 ]
st
s.t. ct + gt + K t+1 = F (Kt , Lt ) + (1 − δ)Kt
• This is like a standard Neoclassical growth model except:
– We have modified the preferences from u to W . We’ll see what that implies.
– The initial period is special (this will be a source of time inconsistency)
• For t 6= 0, FOCs are:
βWc (ct , Lt ) − γ t = 0
βWL (ct , Lt ) + γ t FL Kt , Lt , st , t = 0
−γ t + γt+1 [FK (K t+1 , Lt+1 ) + (1 − δ)] = 0
• Derive two conditions from this
– This is like a planner’s problem of the Neoclassical Growth Model with elastic labour
supply, except with distorted preferences
1
Where does τ0k enter the program? In the definition of R0 .
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• Intratemporal:
−
WL (ct , Lt )
= FL (Kt , Lt )
Wc (ct , Lt )
(11)
• Intertemporal:
∗
Wc (ct , Lt ) = βWc (ct+1 , Lt+1 ) Rt+1
(12)
where
Rt∗ ≡ FK (Kt , Lt ) + (1 − δ)
is the before-tax gross social return on capital
• Recall, from household problem, using (8) and (7):
uL (ct , Lt )
FL (Kt , Lt ) 1 − τ l t = −
uc (ct , Lt )
so using (11) we solve out for the optimal labor tax:
τ l∗ = 1 −
uL (ct , Lt ) Wc (ct , Lt )
WL (ct , Lt ) uc (ct , Lt )
(13)
• Also recall the Euler equation from the household problem, using (9):
uc (ct , Lt ) = βuc (ct+1 , Lt+1 ) R(st+1 )
(14)
• Equations (14) and (12) imply:
∗
Rt+1 = Rt+1
Wc (ct+1 , Lt+1 ) uc (ct , Lt )
uc (ct+1 , Lt+1 ) Wc (ct , Lt )
• Taxes ⇔ difference between true preferences and distorted pseudo-preferences.
4.1
Zero Capital Taxation in the Steady State
Proposition 2. Suppose there is a steady state. Then in the steady state τ k = 0 is optimal.
Proof. Impose steady state on (15)
R(st+1 ) = R∗ (st+1 )
which is achieved with τ k = 0.
This result is due to (Chamley [1986]) and (Judd [1985]).
What is going on?
8
(15)
Econ 210, Fall 2013
Pablo Kurlat
• Uniform commodity taxation [Atkinson and Stiglitz, 1972].
• Steady state capital supply perfectly elastic:
1 = βR
= β 1 + (1 − τ k )(FK − δ)
4.2
Capital Taxation Outside Steady State
• Special case:
u(c, L) =
c1−σ
− v(L)
1−σ
• Then
c1−σ
− v(L) + µ c−σ c − v 0 (L)L
1−σ
1
+ µ c1−σ − [v(L) + µv 0 (L)L]
=
1−σ
W (c, L) =
so
Wc = (1 + µ (1 − σ)) c−σ
= (1 + µ (1 − σ)) uc
Wc
= 1 + µ (1 − σ)
uc
• Therefore (15) reduces to:
R(st+1 ) = R∗ (st+1 )
so for these preferences τ k = 0 is optimal even outside of steady state, for every period other
than the first one.
4.3
Tax smoothing
• Special case:
Lγ
v(L) = α
γ
1
1
1−σ
W (c, L) =
+µ c
−α
+ µ Lγ
1−σ
γ
• Same functional form as u(·) but with more weight on the disutility of labour (as long as
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γ > 1 − σ). Furthermore
Wc
= 1 + µ (1 − σ)
uc
WL
= 1 + µγ
uL
so (13) becomes
τ∗ = 1 −
1 + µ (1 − σ)
1 + µγ
• Tax rate constant over time
• With uncertainty (and complete markets), it’s optimal to perfectly smooth tax rates across
states of the world as well as across time.
• Outside special case, Chari et al. [1994] find that the optimal labour income tax has tiny
fluctuations. Aiyagari et al. [2002] study the case with uncertainty and no state-contingent
debt.
5
Initial period taxation and time inconsistency
5.1
Capital-income taxation
Return to government’s problem and assume τ0k can be chosen freely:
max
ct ,K t+1 ,τ0k
X
β t W (ct , Lt ) − µuc (c0 , L0 ) R0 B0 + 1 + (1 − τ0k )(r0K − δ) K0
st
s.t. ct + gt + K t+1 = F (Kt , Lt ) + (1 − δ)Kt
FOC w.r.t. τ0k :
−µuc (c0 , L0 ) (r0K − δ)K0 = 0
⇒µ=0
• Tax initial capital until that is enough to pay for all government expenditure forever
• May require τ0k > 1
• Replicable with consumption tax, so no consumption tax is w.l.o.g. only if τ0k is allowed
• Achieve first-best allocation
• Non-distortionary, so no time inconsistency problem arises
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• Suppose we impose τ k ≤ τ̄ and this is not enough to satisfy government budget constraint.
Then τ0k = τ̄ is optimal, but optimal plan is not time consistent
5.2
Initial-period allocations
Now look at the choice of c0 . FOC:
Wc (c0 , L0 ) − µucc (c0 , L0 ) R0 B0 + 1 + (1 − τ0k )(r0K − δ) K0 − γ0 = 0
• These are different from the t 6= 0 FOCs as long as R0 B0 + 1 + (1 − τ0k )(r0K − δ) K0 6= 0,
i.e. as long as the household has nonzero wealth. What is the meaning of the extra term
−µucc (c0 , L0 )?
↑ c0
⇒ ↓ uc (c0 , L0 )
⇒ ↑ pt =
β t uc (ct ,Lt )
uc (c0 ,L0 )
⇒ Increase NPV of future revenue
• Lowering interest rates as a way of defaulting on debt
• Another source of time inconsistency
• Issuing debt of different maturities can eliminate this source of time inconsistency [Lucas
and Stokey, 1983].
6
Lump sum taxation and inequality [Werning, 2007]
• Policies that try to imitate the missing lump sum tax
• Why not just use a lump-sum tax?
• Here: introduce a lump sum tax and model inequality as (possibly) a reason not to use it
• (More general [Mirrlees, 1971] approach: taxes as a nonlinear function of past and present
labour and capital income)
• NPV of lump sum tax is T (w.l.o.g. collected at time zero)
• Heterogeneity:
– Utility functions ui (c, L)
– Initial endowments k0i and B0i
• Example: skill/productivity heterogeneity
L
u (c, L) = u c, i
θ
i
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Pablo Kurlat
No distortion given aggregate quantities
• Aggregates:
Lt =
X
ct =
X
π i Lit
i
π i ci t
i
• Since all households face the same after-tax prices, the intratemporal and intertemporal
conditions:
ui (ci , Li )
wt 1 − τ l t = − Li it it
uc (ct , Lt )
t
β Pr t uic (cit , Lit )
pt =
uic (ci0 , Li0 )
hold for all i, so marginal rates of substitution are equated across workers.
• This isn’t true for general nonlinear taxes, where different households face different marginal
tax rates and hence different after-tax prices. Here all households face the same marginal
tax rates.
• Given aggregates ct , Lt , the assignment of labour and consumption to individuals is efficient.
• No distortion on who gets to work/consume at each date, only in total amounts
• ⇒ ∃ weights ϕ such that, cit , Lit solve:
um (ct , Lt , ϕ) ≡ max
i i
c ,L
s.t.
X
X
π i ϕi ui (ci , Li )
π i Li = Lt
i
X
π i ci = ct
i
• (With the same weights for every period)
• Derive implementability condition:
– For any weights ϕ, we have a pseudo-utility function um (ct , Lt , ϕ)
12
(16)
i
(17)
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Pablo Kurlat
– In any competitive equilibrium, the FOC
um
L (ct , Lt , ϕ)
wt 1 − τ l t = − m
uc (ct , Lt , ϕ)
t m
β u (ct , Lt , ϕ)
pt = mc
uc (c0 , L0 , ϕ)
must hold
– Use these conditions to solve out for prices in the budget constraint of each household,
just like we did in the representative agent case
– Given aggregates and weights, individual consumption and labour are the solution to
program (16). Denote them by ci (c, L, ϕ) and Li (c, L, ϕ)
• The implementability condition for household i is:
X
i
m
i
β t um
c (ct , Lt , ϕ) c (ct , Lt , ϕ) + uL (ct , Lt , ϕ) L (ct , Lt , ϕ)
(18)
t
i
i
= um
c (c0 , L0 , ϕ) R0 B0 + K0 − T
• No prices, no taxes, no individual consumption and labour
• Just aggregates and weights
• (18) and the resource constraint are necessary and sufficient for a competitive equilibrium:
1. Given aggregate allocation and weights, find prices and taxes to fit FOCs
2. Household budget holds ⇒ government budget holds
6.2
The optimal taxation problem
• Solve:
"
max
ct ,Lt ,K t ,ϕ,τ0k ,T
X
t
βt
X
#
π i λi ui ci (ct , Lt , ϕ) , Li (ct , Lt , ϕ)
i
s.t. ct + gt + K t = F (Kt , Lt ) + (1 − δ)Kt
X
t
i
m
i
β um
c (ct , Lt , ϕ) c (ct , Lt , ϕ) + uL (ct , Lt , ϕ) L (ct , Lt , ϕ)
t
i
i
= um
c (c0 , L0 , ϕ) R0 B0 + K0 − T
∀i
• λi are the true Pareto weights on each individual (how much the social planner truly loves
them).
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• In general these weights do not coincide with the weights ϕ. Givent the available tax instruments, the planner cannot fully control how the market weights different households.
• Let π i µi be the multiplier on household i’s implementability constraint. Define:
"
W (c, L, ϕ) =
X
πi
i
λi ui (ci (c, L, ϕ) , Li (c, L, ϕ))
i
m
i
+µi [um
c (c, L, ϕ) c (c, L, ϕ) + uL (c, L, ϕ) L (c, L, ϕ)]
#
• Program becomes
max
X
ct ,Lt ,K t ,ϕ,τ0k ,T
β t W (ct , Lt , ϕ) − um
c (c0 , L0 , ϕ)
st
X
π i µi R0 B0 + 1 + (1 − τ0k )(r0K − δ) K0 − T
i
(19)
s.t. ct + gt + K t = F (Kt , Lt ) + (1 − δ)Kt
• For t 6= 0, same structure as the one with no inequality and no lump-sum taxes.
• If we add constraint T = 0 and assume ui (·) = u (·) ∀i and λi = 1∀i we are back to standard
Ramsey problem.
6.3
Results that carry over from representative agent case
• Same FOCs for optimality:
τ
l∗
Rt+1
(ct , Lt , ϕ) WC (ct , Lt )
um
=1− L
WL (ct , Lt ) um
c (ct , Lt , ϕ)
Wc (ct+1 , Lt+1 ) um
c (ct , Lt , ϕ)
∗
= Rt+1
m
uc (ct+1 , Lt+1 , ϕ) Wc (ct , Lt )
• Zero capital taxation in steady state result goes through immediately
• W function defined differently than before. Similar structure, but summing over heterogeneous individuals.
• Special case:
L γ
c1−σ
θi
u (c, L) =
−α
1−σ
γ
i
Proposition 3. In this special case, if the distribution of skills is constant over time (i.e. θi is
constant for each i), then tax rates should be constant
Proof.
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1. Individual consumption and labour supply levels depend linearly on aggregate quantities. This
comes from solving the static assignment problem:

m
u (c, L, ϕ) ≡ max
i i
c ,L
s.t.
X
X
i
i γ 
i 1−σ
(c )
−α
πϕ 
1−σ
i
i
L
θi
γ

π i Li = L
(20)
(21)
i
X
π i ci = c
i
with FOC:
−π i ϕi α Li
γ−1
θi
−γ
+ π i ηL = 0
−σ
− π i ηc = 0
π i ϕ i ci
Solving
1
ci = P
(ϕi ) σ
1
c
(22)
i
i σ
i π (ϕ )
γ
1
i
L =P
(ϕi ) 1−γ (θi ) γ−1
1
γ
L
(23)
i
i 1−γ (θ i ) γ−1
i π (ϕ )
2. This implies that both um (c, L, ϕ) and W (c, L, ϕ) have the same functional form as the
original utility function, i.e.
c1−σ
Lγ
− Φm
L
1−σ
γ
1−σ
c
Lγ
W (c, L, ϕ) = ΦW
− ΦW
c
L
1−σ
γ
um (c, L, ϕ) = Φm
c
m
W
W
for some constants Φm
c , ΦL , Φc and ΦL which depend on weights ϕ. This comes from
replacing (22) and (23) into (20)
3. Then the proof for the representative-agent case applies immediately
• Tax rates should vary if the distribution of skills varies.
• Equate marginal cost of distortions from redistribution over time.
• Government debt becomes indeterminate (adjust the timing of the lump-sum tax).
• Analogy with Ramsey: W (·) is a reweighting of um (·).
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– In the Ramsey problem, it reweights towards more disutility of labour, because that
reweighting is the way to extract revenue
– In this problem, whether it reweights towards labour or consumption depends on the
direction of desired redistribution.
6.4
Lump-sum tax, initial period taxation and time inconsitency
FOC w.r.t. T :
X
π i µi = 0
i
• With only one type, implementability constraint does not bind
• µi will be positive for those types we want to redistribute away from and negative for those
types you want to redistribute towards
• In general, the term involving T just washes out of program (19)
• FOC w.r.t. τ0k :
X
π i µi K0i = 0
i
• If capital holdings at zero are equal, no need to tax initial capital
• If the types we want to redistribute away from have more capital, then planner’s objective
is increasing in τ0k : increase the tax on initial capital until this is no longer the case
• Redistribution without distortion rather than revenue without distortion
• Does this create a time inconsistency problem? It depends on who (if anyone) accumulates
more assets along equilibrium path
• FOCs for time-zero consumption:
Wc (c0 , L0 , ϕ) − um
cc (c0 , L0 , ϕ)
X
π i µi R0 B0i + R0 K0i − T − γ0 = 0
i
Using FOCs for T and τ0k :
Wc (c0 , L0 , ϕ) − um
cc (c0 , L0 , ϕ)
X
π i µi R0 B0i − γ0 = 0
i
• As before manipulating the time-zero marginal utility of consumption is a way to change the
NPV of future taxes.
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• This is useful if the planner wants to redistribute across households with different initial
holdings of government bonds.
• If government bonds can be taxed directly, then the FOC for taxing them is
X
π i µi B0i = 0
i
and the FOC for c0 is the same as for future periods.
6.5
Is government debt a problem?
• In the standard Ramsey model, yes! The derivative of the planner’s objective with respect
to B0 is
−µuc (c0 , L0 )
because higher government debt requires collecting higher distortionary taxes
• In the model with inequality, it depends! The derivative of the planner’s objective with
respect to B0i is
i i
−um
c (c0 , L0 , ϕ) π µ
so initial government debt owed to individual i is good or bad depending on whether individual i is someone we want to redistribute away from (µi positive, which makes this debt bad)
or towards (µi negative, which makes this debt good). If the government owes one dollar to
each household in the economy, then the change in the planner’s objective is
−um
c (c0 , L0 , ϕ)
X
π i µi = 0
i
so debt is neither good nor bad (it can be undone with a lump-sum tax)
• A public debt problem is really a wealth distribution problem!
• (With an open economy the issue is different: if debt is owed to foreigners it affects the
resource constraint)
7
Nonlinear taxation. The Mirrlees [1971] approach
• Why constrain the tax instruments to τ i = τ k rk − δ K i + τ l wLi + T ?
• In general:
τi = τ
rk − δ K i , wLi
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Econ 210, Fall 2013
Pablo Kurlat
• Or it could also be dependent on past income
• Or on anything else that the government can observe
• Common assumption: the government can observe labour income and capital income but
nothing else. In particular, the government cannot observe ability
– Tax evasion?
– Hidden income?
• What is the set of allocations the government can implement? Use revelation principle!
7.1
Simple static example with two types
• Problem
LL
LH
+ πL λL u (cL ) − v
max πH λH u (cH ) − v
ci ,Li
θ
θ
H L
LH
LL
s.t. u (cH ) − v
≥ u (cL ) − v
θH
θH
πH cH + πL cL + g ≤ πH LH + πL LL
• πi are population proportions
• λi are Pareto weights: λi = 1 is utilitarian case
• Output is just linear in the total effective labour supply
• The government asks each agent to declare their type and then tells them how much to
produce and gives them consumption
• Indifference curves in (L, c) space, single-crossing
• FOC:
– For H types:
(πH λH + µ) u0 (cH ) − ηπH = 0
1 0 LH
− (πH λH + µ) v
+ ηπH = 0
θH
θH
v 0 LθHH
⇒ 0
= θH
u (cH )
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Econ 210, Fall 2013
Pablo Kurlat
– Equate the marginal rate of substitution to the marginal product of labour
– No distortion!
– “No distortion at the top”. You only distort the allocations for the “low” types (i.e.
those that agents want to pretend they are).
– General feature of mechanism-design problems with bounded support
– For L types:
− (πL λL )
1 0
v
θL
(πL λL − µ) u0 (cL ) − ηπL = 0
LL
1 0 LL
+µ v
+ ηπL = 0
θL
θH
θH
LL
θH L
v0 θ L
L
v0
πL λ L − µ
θL u0 (cL )
⇒ =
πL λ L − µ
v 0 LθLL
0 LL
v θL
⇒ 0
< θL
u (cL )
• How do we know that
>1
LL
θH L
v0 θ L
L
v0
θL
θH
θL
θH
< 1?
– Draw the indifference curve for both types with c in the horizontal axis and L on the
vertical axis
– (They are upward sloping)
– The slope is given by
u0 (c)θ
v0 ( L
θ)
– At any point c, L the slope must be steeper for the H type because he has a relatively
lower marginal disutility of labour and the same marginal utility of consumption (“single
crossing”)
– Applying this reasoning to point cL , LL implies
LL
θH L
v0 θ L
L
v0
• Implement this by some tax function τ (L)
• Nonlinear income tax, with marginal tax rate τ 0 (L)
• MRS<MPL needs a positive marginal tax
v0
LL
θL
u0 (cL )
= θL (1 − τ 0 (LL ))
19
θL
θH
< 1.
Econ 210, Fall 2013
Pablo Kurlat
so
πL λ L − µ
τ 0 (LL ) = 1 −
πL λ L −
v0
LL
>0
µ 0 LθHL θθHL
v
θL
• Low types face a positive marginal tax rate
• With more types, similar reasoning pins down the marginal tax rates for any L
References
S. Rao Aiyagari, Albert Marcet, Thomas J. Sargent, and Juha Seppala. Optimal taxation without
state-contingent debt. Journal of Political Economy, 110(6):1220–1254, December 2002.
Anthony B. Atkinson and Joseph E. Stiglitz. The structure of indirect taxation and economic
efficiency. Journal of Public Economics, 1(1):97–119, April 1972.
Christophe Chamley. Optimal taxation of capital income in general equilibrium with infinite lives.
Econometrica, 54(3):607–22, May 1986.
V V Chari, Lawrence J Christiano, and Patrick J Kehoe. Optimal fiscal policy in a business cycle
model. Journal of Political Economy, 102(4):617–52, August 1994.
Kenneth L. Judd. Redistributive taxation in a simple perfect foresight model. Journal of Public
Economics, 28(1):59–83, October 1985.
Robert Jr. Lucas and Nancy L. Stokey. Optimal fiscal and monetary policy in an economy without
capital. Journal of Monetary Economics, 12(1):55–93, 1983.
James A Mirrlees. An exploration in the theory of optimum income taxation. Review of Economic
Studies, 38(114):175–208, April 1971.
Iván Werning. Optimal fiscal policy with redistribution. The Quarterly Journal of Economics, 122
(3):925–967, 08 2007.
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