Federal Reserve Bank of Chicago
Redistribution, Taxes, and the
Median Voter
Marco Bassetto and Jess Benhabib
WP 2006-02
Redistribution, Taxes, and the Median Voter∗
Marco Bassetto
Federal Reserve Bank of Chicago
University of Minnesota and NBER
Jess Benhabib
New York University
Abstract
We study a simple model of production, accumulation, and redistribution, where agents
are heterogeneous in their initial wealth, and a sequence of redistributive tax rates is voted
upon. Though the policy is infinite-dimensional, we prove that a median voter theorem
holds if households have identical, Gorman aggregable preferences; furthermore, the tax
policy preferred by the median voter has the “bang-bang” property.
JEL numbers: H21, H23
∗
We thank Michele Boldrin, Adam Przeworski, and anonymous referee for comments and Onur Ozgur for
technical assistance. Marco Bassetto thanks the NSF and the Sloan Foundation for financial support. The views
expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Chicago, the
Federal Reserve System, or NSF.
1
1
Introduction
We study a simple model of production, accumulation, and redistribution, where agents are
heterogeneous in their initial wealth, and where a sequence of redistributive tax rates are chosen
through voting. Decisions to save are endogenous, which means that they depend on the time
profile of future tax rates.
Since the tax rate need not be constant, we encounter all the richness and all the difficulty of
the optimal tax literature (see for example Chamley (1985), (1986), Judd (1985), Benhabib and
Rustichini (1997), Chari and Kehoe (1999), Atkeson, Chari and Kehoe (1999)). Furthermore,
since agents vote over a sequence of tax rates, the usual single-peakedness assumption required
for the median voter theorem cannot be used. While poorer households will tend to favor more
redistribution through capital-income taxes than richer households, the specific path preferred by
each household may vary in complicated ways that depend on the distribution of local elasticities
of intertemporal substitution.
The contribution of our paper is twofold. First, we prove that, when preferences are identical
and Gorman aggregable, a Condorcet winner exists. This is achieved by showing that preferences
satisfy an order restriction, as discussed in Rothstein (1990,1991) and in Gans and Smart (1996).
This technique can have a wider application in environments where voting occurs over sequences
of policies but heterogeneity among voters is restricted to a single dimension. Second, we prove
that the policy chosen by majority voting has the “bang-bang” property: capital income taxes
remain at the upper bound until they drop to 0, with at most one period in between. This
generalizes to heterogeneous households a result first proven by Chamley (1986) in the context
of a representative agent. When redistributive considerations are strong and the government
ability to tax capital is limited, it is possible that the median voter prefers to keep the tax rate
at its maximum allowed value in all periods, as we show through an example.
The “bang-bang” property collapses a multidimensional policy vector into a single choice
over the optimal stopping time, with poorer households favoring maximal capital-income taxes
for a longer period than rich ones. Gorman aggregation, however, delivers a stronger result, as
a median voter result applies to all pairwise comparisons of policy sequences.
2
2
The Model
We have a production economy where output yt at time t is produced by competitive firms using
capital kt and labor lt according to the production function
yt = F (kt , lt ),
(1)
with F linearly homogeneous. Without loss of generality, capital depreciates fully in each period.1
There is a continuum of agents of unit measure, with agents indexed by i. In the initial period
0, they each own a wealth level, W0i (made of capital and government debt). In each period,
the government levies nonnegative proportional taxes on labor income ν t and capital income τ t ,
subject to an exogenous upper bound τ̄ .2 The government uses tax receipts and one-period debt
to pay for spending on a public good at an exogenous rate {gt }∞
t=0 , and to finance a lump-sum
transfer Tt that is used for redistribution.3
Household preferences are given by
∞
β t u cit
t=0
where cit is period t consumption by agent i. We will assume that household utilities are restricted to satisfy Gorman aggregation, that is to have linear Engel curves. These preferences
1
2
Any undepreciated capital can be accounted for by adding (1 − δ)kt to the definition of F .
The upper bound could be justified by the presence of a “black market technology” that allows households
to shield their income from observation by the tax collector at a proportional cost τ̄ . The zero lower bound will
not be binding for the distribution of wealth that we will consider. Also, notice that we assumed the tax rate to
hit both principal and income from capital. This can be changed with no effect on the results.
3
Bassetto (2005) proves that the full specification of a government strategy is needed to ensure that a given
policy choice does not give rise to multiple equilibria. We assume here that the government will adjust government
spending in response to unanticipated shortfalls/excesses in tax or borrowing revenues in any given period. This
ensures a unique equilibrium, given tax rates. While government spending is allowed to vary, voters are restricted
to choose policies such that the exogenous target {gt }∞
t=0 is achieved at the equilibrium, which would be justified
if all households receive a sufficiently large disutility from deviations from this target. Our analysis could be
extended to account for endogenous spending.
3
are exhausted by the exponential and power classes (see Pollak, 1971) which coincide with the
HARA class of utility functions.
Assumption 1 Household preferences are given by4
1−σ
σ
A
u(c) =
.
c+B
1−σ σ
(2)
Each household is endowed with 1 unit of labor, which it supplies inelastically. Wti is the
wealth of household i at time t, consisting of capital and maturing bonds. The period-by-period
household budget constraint is
i
yti = (1 − τ t ) rt Wti + (1 − ν t ) wt + Tt ≥ cit + Wt+1
,
(3)
where rt is the gross return to capital and wt is the wage at time t. We have imposed the no
arbitrage condition by stipulating that the net return on bonds and capital are equal. We also
assume that households cannot run Ponzi schemes:
t+1
i
lim ( (rs (1 − τ s ))−1 )Wt+1
≥0
t→∞
(4)
s=1
The household budget constraint in present value form is:
∞
t=0
cit
t
−1
(rs (1 − τ s ))
=
W0i
s=0
+
∞
(Tt + wt (1 − ν t ))
t=0
t
(rs (1 − τ s ))−1
(5)
s=0
For convenience we define prices qt :
qt = β −t
t
(rs )−1
(6)
s=1
Rather than working with the sequence {τ t }∞
t=0 , it is more convenient to work with the following
transformation, which is one to one over the relevant domain:
t
1 + θt :=
(1 − τ s )−1
(7)
s=0
4
The HARA class reduces to logarithmic utility for σ = 1, to quadratic utility for σ = −1, to CRRA utility
for σ > 0 and B = 0, to linear utility for σ = 0, and to exponential utility for σ → ∞ and B > 0.
4
We can write (5) as:
∞
β t cit qt
W0i
(1 + θt ) =
+
t=0
∞
(Tt + wt (1 − ν t )) β t qt (1 + θt )
(8)
t=0
Let cs and bs represent the aggregate levels of consumption and bonds at time s.
Definition 1 A competitive equilibrium is
∞
cs , ks , bs , τ s , ν s , Ts , rs, ws , cis , Wsi i∈(0,1)
s=0
that satisfies
1.
cis , Wsi i∈(0,1)
∞
s=0
(9)
maximize household utilities subject to (3) and (4);
2. Factor prices equal their marginal products:
rt = Fk (kt , 1) , wt = Fl (kt , 1) ;
3. Markets clear
cit di
= ct ,
Wti di = kt + bt ;
4. The government budget satisfies:
a) τ t rt (kt + bt ) + ν t wt + bt+1 = rt bt + gt + Tt
t+1
b) lim ( (rs (1 − τ s ))−1 )bt+1 = 0
t→∞
(10)
s=1
Let the sequence {gs }∞
s=0 represent the exogenous government expenditures. With no assumptions on the sequence, it would of course be possible that some gs > f s (f s−1 (...f 0 (k0 ))
where f n (kn ) = F (kn , 1) , so that the feasibility condition (11) in the Theorem below cannot
be satisified; however, a competitive equilibrium will exist if we assume that {gs }∞
s=0 is not too
high. We assume further that this sequence is sufficiently small that no household faces a budget
5
constraint with negative resources in any of the competitive equilibria analyzed below.5 First, we
characterize properties of the competitive equilibria that will be used in proving the subsequent
theorems.
Theorem 1 For any sequence {cs , ks }∞
s=0 , there exists a competitive equilibrium
∞
cs , ks , bs , τ s , ν s , Ts , rs, ws , cis , Wsi i∈(0,1)
s=0
if and only if the sequence satisfies
1.
kt+1 + ct + gt = F (kt , 1)
2.
(Fk (kt+1 , 1) (1 − τ̄ ))−1 ≤
β t u (ct+1 )
≤ (Fk (kt+1 , 1))−1
u (ct )
(11)
(12)
Proof. First, we prove that conditions 1 and 2 are sufficient. We set rt = Fk (kt , 1) , wt =
Fl (kt , 1) . Taxes on capital satisfy
−1
(1 − τ t+1 )
β t rt+1 u (ct+1 )
1 + θt+1
=
.
=
u (ct )
1 + θt
(13)
In present value form the government budget constraint is:
∞
t
(wt ν t − Tt ) β qt (1 + θt ) =
t=0
∞
(gt − τ t rt kt ) β t qt (1 + θt ) + b0
t=0
Pick any sequence {ν t , Tt , bt+1 }∞
that satisfies the above, as well as (10).6 This sequence is not
0
unique because of Ricardian equivalence.
5
The analysis could be extended to the case in which some households are insolvent, but this would require
assumptions about the consequences of insolvency. Notice also that we abstract from the nonnegativity constraint
on consumption; if σ > 0 and B ≤ 0, the constraint will never be binding.
∞
6
It is possible that for a given {gs }s=0 with large elements some of the lump-sum transfers Tt will be negative,
since we have not explicitly constrained them to be non-negative. However, as is clear from Theorem 3 below,
a median voter with wealth below the mean will not prefer a tax sequence that relies solely on lump-sum taxes
because of distribution considerations.
6
The household first order condition, iterating (13), is
A
cit
Aσ i
A σ c0
A
+B
+B
−σ
−σ
−σ
A Aσ ct + B
qt (1 + θt )
=
= A
−σ
1 + θ0
A σ c0 + B
This implies
A i
ct + B = α i
σ
A
ct + B
σ
(14)
(15)
If αi is set such that the present value budget constraint (5) holds at the given prices and taxes,
the resulting sequence solves the household’s optimization problem.
We now have to show that cit di = ct , which will hold if αi di = 1. Integrating the present
value budget constraints over households we have:
∞
t
β qt (1 + θt )
t=0
W0 +
∞
σB (αi − 1)
α ct +
A
i
di =
(16)
t
(Tt + wt (1 − ν t )) β qt (1 + θt )
t=0
From the feasibility constraint we have ct + gt = F (kt , 1) − kt+1 , which implies
∞
(ct + gt ) β t qt (1 + θt ) =
t=0
∞
F (kt , 1) β t qt (1 + θt ) −
t=0
∞
kt+1 β t qt (1 + θt )
t=0
Using the zero-profit condition of the firms we have:
∞
t
(ct + gt ) β qt (1 + θt ) = k0 +
t=0
∞
(wt + τ t rt kt ) β t qt (1 + θt )
t=0
Using (10) this is equivalent to:
∞
t=0
t
ct β qt (1 + θt ) = W0 +
∞
(Tt + wt (1 − ν t )) β t qt (1 + θt )
(17)
t=0
For (17) and (16) to hold simultaneously we must have
αi di = 1, which implies market clearing.
To prove necessity, the feasibility condition 1 is implied by market clearing and the household
and government budget constraints; condition 2 is implied by the first-order conditions of the
household problem (14) and the bounds on capital-income tax rates.
7
3
The Median Voter Theorem
In this section we prove that the median voter theorem will hold in our setting. We assume
that the sequence of taxes and transfers is set by voting at time 0. Since agents vote over an
infinite sequence of tax rates and transfers, the usual single-peakedness assumption required for
the median voter theorem cannot be used. Our proof will instead rely on the following two
observations:
1. Households differ from each other along a single dimension, i.e., their initial capital holdings.
2. Any change in tax rates affects households in two ways: by the redistribution of wealth that
it implies, and by the distortion in after-tax prices that it generates. Gorman aggregation
and the fact that all households have the same discount factor imply that the distortion in
after-tax prices has a proportional effect on all households, independent of wealth. Thus,
all households trade off a single, common measure of distortions against the degree of
redistribution engineered by the distortion. Households of different wealth will disagree
on the optimal point along this trade-off, but their disagreement will naturally be ordered
according to their initial wealth level.
To prove the theorem we first establish the following Lemma:
Lemma 1 For each household i there exists a function G : R4 → R such that the utility of the
household in a competitive equilibrium is G (V, c0 , τ 0 , W0i − W0 ) where V, the utility of the agent
t
with average wealth, is ∞
t=0 β u (ct ). Also,
∂G (V, c0 , τ 0 , W0i − W0 )
∂G (V, c0 , τ 0 , W0i − W0 )
= sign
=
sign
∂c0
∂τ 0
sign W0 − W0i .
Proof. Subtracting the average budget constraint from the budget constraint of household i
and substituting (14) we obtain:
−σ
−σ
∞
i
A
A
i
t
β ct − ct
= r0 (1 − τ 0 ) W0 − W0
ct + B
c0 + B
σ
σ
t=0
8
Using (15) we obtain:
1−σ
−σ
A
i
A
= r0 (1 − τ 0 ) W0 − W0
ct + B
c0 + B
σ
σ
−σ
i
A
1−σ
i
α −1
V = r0 (1 − τ 0 ) W0 − W0
c0 + B
A
σ
∞
σ
α −1
βt
A t=0
i
Ar0 (1 − τ 0 ) (W0i − W0 )
α =1+
V (1 − σ)
i
A
c
σ 0
+B
−σ
Therefore the utility attained by household i is:
1−σ
G V, c0 , τ 0 , W0i − W0 = αi
V =
−σ 1−σ
Ar0 (1 − τ 0 ) (W0i − W0 ) Aσ c0 + B
1+
V
V (1 − σ)
(18)
The partial derivatives follow trivially.
Theorem 2 The tax sequence preferred by the household with median wealth is a Condorcet
winner.7
Proof. We will use an order restriction to prove the theorem. Consider two competitive
∞
equilibrium sequences {ct }∞
t=0 and {ĉt }t=0 , and two initial tax rates τ 0 and τ̂ 0 . Define V :=
1−σ
1−σ
∞ σ A
∞ σ A
c
+
B
and
V̂
:=
ĉ
+
B
.
t
t
t=0 1−σ σ
t=0 1−σ σ
Construct the set of households that (weakly) prefer the competitive equilibrium associated
∞
with ({ct }∞
t=0 , τ 0 ) to the one associated with ({ĉt }t=0 , τ̂ 0 ):
A
−σ 1−σ
i
(1
−
τ
)
(W
−
W
)
c
+
B
Ar
0
0
0
0
σ 0
H := W0i : 1 +
V ≥
V (1 − σ)
−σ 1−σ Ar0 (1 − τ̂ 0 ) (W0i − W0 ) Aσ ĉ0 + B
V̂
1+
V̂ (1 − σ)
7
(19)
An alternative proof of this result which only works for CRRA preferences and linear technology is in Benhabib
and Przeworski (2006).
9
Conversely, let Ĥ be the set of households that (weakly) prefer the other equilibrium:
A
−σ 1−σ
i
(1
−
τ
)
(W
−
W
)
c
+
B
Ar
0
0
0
0
0
σ
Ĥ := W0i : 1 +
V ≤
V (1 − σ)
−σ 1−σ Ar0 (1 − τ̂ 0 ) (W0i − W0 ) Aσ ĉ0 + B
V̂
1+
V̂ (1 − σ)
(20)
We need to prove that both H and Ĥ are convex, independently of the choice of sequences.
To do so, we consider the ratio of the utility for a household with initial capital W0i in the two
equilibria, and we take the derivative of its logarithm, which is
−σ
(1 − σ) 1+
Ar0 (1−τ 0 )( A
c +B )
σ 0
V (1−σ)
−
−σ −1
Ar0 (1−τ 0 )(W0i −W0 )( A
c +B )
σ 0
V (1−σ)
−σ
Ar0 (1−τ̂ 0 )( A
ĉ +B )
σ 0
V̂ (1−σ)
−σ
1+
Ar0 (1−τ̂ 0 )(W0i −W0 )( A
ĉ +B )
σ 0
−1
V̂ (1−σ)
The sign of the derivative is independent of W0i . This proves the convexity of H and Ĥ, and the
theorem.
4
The Taxes Preferred by the Median Voter
We assume that the median voter’s wealth, W0m , is below the mean.8 We show in the theorem
below that the capital-income taxes preferred by the median voter are decreasing over time and
have have the “bang-bang” property: they are either at their maximum level or at 0 in all periods
except at most one. The median voter will always prefer maximal taxes on capital in period 0
and strictly positive taxes in period 1, which rules out no taxes as the voting equilibrium. We
provide an example where the preferred tax rate remains at the upper bound for ever.9
Theorem 3 The capital tax sequence {τ t }∞
0 preferred by the median voter has the bang-bang
property: if τ t < τ̄ , then τ s = 0 for s > t.
8
If the median voter has wealth above the mean, then using arguments similar to those below, it can be shown
that he will prefer zero capital taxes forever.
9
This example is not inconsistent with Judd (1985): Judd proves that the taxes preferred by any agent in the
economy converge to 0 if the economy is at a steady state and the equilibrium is interior.
10
Proof. From Lemma 1, the initial tax will be τ̄ if the wealth of the median voter is below
the mean.
Case 1: At the allocation preferred by the median voter G (V, c0 , τ 0 , W0m − W0 ) is increasing
in V .
To derive implications for the entire sequence of taxes, consider first the consumption and
capital sequence that maximizes V. This will solve
max
{ct }∞
t=0
∞
t=0
σ
β
1−σ
t
A
ct + B
σ
1−σ
subject to (11) and (12). The solution involves the Euler equation u (ct−1 ) = βFk (kt , 1) u (ct )
and hence setting θt = 0 (or τ t = 0) for all t ≥ 1: since V is the utility of the household
having mean wealth, this household does not have any incentive to distort the economy. Since
we assumed the government is constrained to nonnegative capital income taxes, this solution is at
a corner; in terms of the allocation, it is at the maximum growth compatible with the constraint
(12), though the constraint is not binding.
Let {c∗t , kt∗ }∞
t=0 be the preferred sequences of aggregate consumption and capital by a household with median wealth within those that satisfy (11) and (12). This allocation will be implemented by a sequence of capital income taxes {τ ∗t }∞
1 .
An equivalent statement for the theorem is , if u (c∗t ) > [βFk (kt+1 , 1) (1 − τ̄ )]u c∗t+1 , then
u (c∗s ) = βFk (ks+1 , 1) u c∗s+1 for all s > t. Suppose this were not true. Then {c∗i }∞
s=t+1 does
not satisfy the first-order conditions for solving
∞
σ
β
max
{cs }∞
1−σ
s=t+1
s=t+1
s
A
ct + B
σ
1−σ
(21)
subject to (11) and (12) from period t + 1 on, given kt+1 . It is thus possible to find an alternative
∞
∗ t
∗∗ ∞
sequence {c∗∗
s }s=t+1 such that ({c }s=0 , {cs }s=t+1 ) satisfies (11) and (12), but such that, for
11
sufficiently small ε > 0,
1−σ
1−σ
∞
A ∗∗
A ∗
σ
s σ
β
=
β
+ ε =⇒
c +B
c +B
1−σ σ t
1−σ σ t
s=t+1
s=t+1
1−σ
1−σ
t
∞
σ
A ∗
A ∗∗
s σ
β
+
c +B
c +B
1−σ σ t
1−σ σ t
s=0
s=t+1
1−σ
∞
A ∗
s σ
c +B
=
β
+ε
1−σ σ t
s=0
∞
s
(22)
The new sequence has the same initial consumption, but a higher value for the utility aggregate
V . By hypothesis we have G (V, c0 , τ 0 , W0m − W0 ) strictly increasing in V. As a consequence, the
new sequence would be preferred by the median household, which is a contradiction.
Case 2: At the allocation preferred by the median voter G (V, c0 , τ 0 , W0m − W0 ) is decreasing
in V . Then we can prove by contradiction that capital income taxes must be at their upper bound
at all periods. Suppose this were not true. Then pick the first period in which the growth rate
of marginal utility is below its lower bound:
[β (1 − τ̄ ) Fk (kN , 1)] <
u (cN −1 )
u (cN )
Then change u (cN −1 ) proportionately by a factor dΨ (this entails raising the capital tax rate
in period N ) and change u (cN ) ...u (cM ) by a corresponding factor dΦ so feasibility remains
satisfied, where M is the first period after N where:
[βFk (kM +1 , 1)] >
u (cM )
u (cM +1 )
The required adjustment in ct is:
u (cN −1 )
dcN −1
= = − σ −1 cN −1 + A−1 B
dΨ
u (cN −1 )
u (ct )
dct
f or t = N, ...M
= = − σ −1 ct + A−1 B
dΦ
u (ct )
The implied change in capital is:
dkN = −dcN −1 = dΨ σ −1 cN −1 + A−1 B
12
dkt = dΨ
t−1
Fk (kj , 1)
t−1 t−1
σ −1 cN −1 + A−1 B + dΦ
Fk (kj , 1) σ −1 cs + A−1 B
s=N j=s+1
j=N
for t = N + 1, ...M + 1. Since dkM +1 = 0,10
−1
0 = dΨ Aσ cN −1 + B + dΦ
M s
(Fk (kj , 1))−1 Aσ −1 cs + B
s=N j=N
From the Euler equation, and the non-negativity of capital income taxes:
−σ −1
−σ
≥ Aσ ct−1 + B
βFk (kt , 1) Aσ −1 ct + B
If dΨ < 0 this implies that
−1
−dΨ Aσ cN −1 + B
1−σ
≥ dΦ
M
1−σ
β s−N +1 Aσ −1 cs + B
s=N
The effect that this change has on the utility index is:
dV
=
M
β s u (cs )dcs
s=N −1
M
1−σ
1−σ
= β N −1 Aσ −1 cN −1 + B
dΨ +
β s Aσ −1 cs + B
dΦ ≤ 0
s=N
The change will be strictly negative, unless u (cN −1 )/u (cN ) = βFk (kN , 1), i.e., unless τ N = 0.
t
If instead τ N = 0, then the unperturbed allocation (cN −1 , . . . , cM ) maximizes M
t=N −1 β u(ct )
subject to (11) and to the initial and terminal values for capital, kN −1 and kM +1 . By strict
concavity of utility, the perturbed allocation has a negative (second-order) effect on V .
Finally we can show that the perturbation constructed above to satisfy (11) also satisfies (12).
The perturbation raises cN −1 and decreases cN through cM , in a way that leaves the marginal
rate of substitution between cj and cj+1 unaffected when N < j < M . So between N − 1 and
N the perturbation raises the capital income tax, which was not at the maximum. Between j
and j + 1, with N < j < M , the marginal rate of substitution is unaffected, but the marginal
productivity of capital is higher due to the higher consumption and lesser accumulation in period
10
If M = ∞, we can take limM →∞
M
j=N
−1
Fk (kj , 1)
dkM +1 = 0.
13
N − 1. Therefore for the competitive equilibrium to hold, we need to raise the capital income
tax from 0 to a positive number, which is feasible. Between periods M and M + 1, we need to
lower the capital income tax and we can do that since it is positive.
Case 3: At the allocation preferred by the median voter G (V, c0 , τ 0 , W0m − W0 ) is stationary
with respect to V . Then we can prove again that capital income taxes must be at their upper
bound at all periods. Suppose this were not true. Then pick the first period in which the growth
rate of marginal utility is below its lower bound:
[β (1 − τ̄ ) Fk (kN , 1)] <
u (cN −1 )
u (cN )
Then we can increase the consumption in periods 0, . . . , N − 1 and decrease consumption in
periods N, . . . , M , where M is the first period after N where:
[βFk (kM +1 , 1)] >
u (cM )
u (cM +1 )
This perturbation can be achieved by raising capital income taxes in period N and lowering them
in period M + 1, while keeping them at τ̄ for t < N and at 0 for periods N + 1 < t < M + 1.
This perturbation yields a first-order increase in initial consumption c0 , which brings about a
first-order benefit on the median voter; if G is stationary with respect to V , any cost from the
distortions imposed on V would only have a higher-order impact. Hence, the perturbation would
be beneficial.
Theorem 4 In period 0, the capital tax preferred by the median voter is τ̄ . In period 1, it is
strictly positive.
Proof. Equation (18) is strictly increasing in τ 0 for W0i < W0 , hence setting τ 0 = τ̄ is
optimal for the median voter. Intuitively, the initial capital-income tax is lump-sum: it achieves
redistribution at no cost in terms of distortions.
To prove that the preferred level of τ 1 is strictly positive, notice that V is maximized by the
sequence τ t = 0, t ≥ 1, and that the constraint τ t ≥ 0 is not binding. Hence, any increase in τ 1
locally has only second-order effects on V , but feasibility and (14) imply that c0 has a first-order
14
increase. Equation (18) will thus increase as τ 1 is perturbed from 0 to a strictly positive number
(holding all future taxes at 0).
While the theorem above rules out no taxes for ever as a possible outcome, it is instead
possible that the median voter will find maximal taxes forever to be its preferred choice, if the
redistribution concern is sufficiently strong. Let the ratio of the initial median wealth to mean
wealth be R =
W0m
.
W0
Corollary 1 If preferences are CRRA (B = 0 and σ > 0 in (2)) and production is linear (y = rk
in (1)), the capital tax preferred by the median voter is τ̄ forever if
1+ 1
σ
1−β r
1−σ
σ
σr (1 − τ̄ ) (R − 1)
−1 ≤ 0,
1
1
1−σ
(1 − τ ) σ r 1 − β σ (r (1 − τ )) σ
which can only happen if σ > 1.11
Proof. Under the CRRA preferences and linear technology
G (V, c0 , τ 0 , W0m − W0 ) is (weakly) decreasing in V if
−σ Ar0 (1 − τ 0 ) (W0i − W0 ) Aσ c0 + B
1+σ
≤0
(1 − σ) V
(23)
Furthermore, it is straightforward to prove that, when σ > 1, ∂ 2 G/∂V 2 < 0, ∂ 2 G/∂c0 ∂V > 0,
and ∂ 2 G/∂V ∂τ 0 > 0. The competitive equilibrium with τ t = τ̄ for all periods t ≥ 0 has
both the lowest value of V and the highest values of c0 and τ 0 among all competitive equilibria.
Hence, it is sufficient to check that ∂G/∂V is strictly negative at this equilibrium
to ensure that it is negative at all possible equilibria. Theorem 3 then implies the desired
For taxes set at τ t = τ̄ , t ≥ 0, the discounted utility and initial consumption of
−1
1
1−σ
and
the agent with average wealth W0 are V = σ (1 − σ)−1 (c0 )1−σ 1 − β σ (r0 (1 − τ̄ )) σ
1−σ
1
c0 =
1 − β σ (r (1 − τ̄ )) σ (1 − τ̄ ) + τ̄ rW0 . Substituting these into (23) we obtain the
result.
Corollary.
11
It can also be proven that the condition in the corollary is necessary; for linear preferences and CRRA utility,
taxes will necessarily converge to 0 if (18) is strictly increasing in V .
15
As an example of the Corollary, consider the case r = 1/β, σ = 2, B = 0 and β = .96, and
no government spending.12 If more than 50% of the population has no capital, maximal taxes
for ever will be the political outcome whenever τ̄ < 2.63%.13
5
Conclusion
In this paper, we established a median voter result for a class of economies in which an entire
sequence of tax rates is chosen once and for all, and we characterized the solution preferred by
the median voter. We proved that each household finds it optimal for the economy to converge
to a steady state, so that Judd’s (1985) result of no capital income taxes in the limit applies,
with the exception of cases in which an upper bound on capital income taxation binds for ever.
Our results can be useful even in environments where government policy is not set once
and for all. First, establishing that the household with median wealth is pivotal in all pairwise
comparisons of policy sequences is a likely useful step in proving existence of a median voter for
dynamic political-economic equilibria where policy choices are made more frequently. Secondly, it
is often the case that governments choose infrequently sequences of tax policies that will remain
in effect for a finite period of time. Our method, based on Rothstein’s (1990,1991) original
insight, can be adapted to prove the existence of a median voter for these multidimensional
choices as well.14
6
References
Atkeson, Andrew , Chari, V. V., and Kehoe, Patrick J. (1999). Taxing Capital Income: A
Bad Idea, Federal Reserve Bank of Minneapolis Quarterly Review, Vol. 23, pp. 3-17.
12
13
A can be normalized to any positive value.
If the tax rate applies only to capital income net of depreciation, rather than to both capital and its income,
i.e., only to (r − 1)k rather than to rk, the corresponding tax rate is 66%.
14
For a related application of Rothstein’s method, see Jack and Lagunoff (2006).
16
Bassetto, Marco (2005). Equilibrium and Government Commitment. Journal of Economic
Theory, vol. 124, pp. 79-105.
Benhabib, Jess and Przeworski, Adam (2006). The Political Economy of Redistribution under
Democracy. Economic Theory, fortcoming. Working paper version (2003):
http://www.econ.nyu.edu/user/benhabib/dgcesdraft87.pdf
Benhabib, Jess and Rustichini, Aldo (1997). Optimal Taxes Without Commitment. Journal
of Economic Theory, vol. 77, pp. 231-259.
Chamley, Christophe (1985). Efficient Taxation in a Stylized Model of Intertemporal General
Equilibrium. International Economic Review, vol. 26, pp. 451-468.
Chamley, Christophe (1986). Optimal Taxation of Capital Income in General Equilibrium
with Infinite Lives. Econometrica, vol. 54, pp. 607-622.
Chari, V. V., and Kehoe, Patrick J. (1999). Optimal Monetary and Fiscal Policy. In: Handbook for Macroeconomics, eds. John Taylor and Michael Woodford, North-Holland, New York,
volume 1C, pp. 1671-1745.
Gans, Joshua S. and Smart, Michael (1996). Majority Voting with Single-Crossing Preferences. Journal of Public Economics, vol. 59, pp. 219-237.
Jack, William and Roger Lagunoff (2006). Dynamic Enfranchisement. Journal of Public
Economics, forthcoming.
Judd, Kenneth (1985). Redistributive Taxation in a Simple Perfect Foresight Model. Journal
of Public Economics, vol. 28, pp. 59-83.
Pollak, Robert A. (1971). Additive Utility Functions and Linear Engel Curves. Review of
Economic Studies, vol. 38, pp. 401-414.
Rothstein, Paul (1990). Order Restricted Preferences and Majority Rule, Social Choice and
Welfare, vol. 7, pp. 331-342.
Rothstein, Paul (1991). Representative Voter Theorems, Public Choice, vol. 72, pp. 193-212.
17
Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
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1
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Working Paper Series (continued)
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Working Paper Series (continued)
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Working Paper Series (continued)
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6