The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Working Paper Series
The Political Economy of Labor
Subsidies
WP 06-09
Marina Azzimonti
University of Iowa
Federal Reserve Bank of Richmond
Eva de Francisco
Towson University
Per Krusell
Princeton University
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
The political economy of labor subsidies
∗
Marina Azzimonti†, Eva de Francisco‡, and Per Krusell§
September, 2006
Federal Reserve Bank of Richmond
Working Paper No. 06-09
Abstract
We explore a political economy model of labor subsidies, extending Meltzer
and Richard’s median voter model to a dynamic setting. We explore only one
source of heterogeneity: initial wealth. As a consequence, given an operative
wealth effect, poorer agents work harder, and if the agent with median wealth
is poorer than average, a politico-economic equilibrium will feature a subsidy to
labor. The dynamic model does not have capital, but it has perfect markets for
borrowing and lending. Because tax rates influence interest rates, another channel
for redistribution appears, since a decrease in current interest rates favors agents
with a negative (below-average) asset position.
By the same token—and as is typically the case in dynamic politico-economic
models with rational agents—the setting features time-inconsistency: the median
voter would like to commit to not manipulating interest rates in the future. Under
commitment, and under the assumption that preferences admit aggregation, we
show that labor subsidies subsist only for one period; after that, subsidies are
zero. That is, under commitment, the median voter takes advantage of the voting
power once and for all. His wealth moves closer to that of the mean (which is zero),
but afterwards he refrains voluntarily from further subsidization. Under lack of
commitment, which we analyze formally by looking at the Markov-perfect (timeconsistent) equilibrium in a game between successive median voters in the same
environment. Instead, subsidies persist—they are constant over time—and are
more distortionary than under commitment. Moreover, in the situation without
commitment, the median voter does not manage to reduce asset inequality, unlike
in the commitment case.
JEL Classification: E61, E62, H11, H29, H41, O23
Keywords: Commitment, Median Voter Theorem, Markov Equilibrium, Time Consistency,
Labor Subsidies, Aggregation.
∗
We wish to thank Atsushi Oshima for excellent research assistance.
University of Iowa and Federal Reserve Bank of Richmond. Email: marina-azzimonti@uiowa.edu.
‡
Towson University. Email: EDefrancisco@towson.edu.
§
Princeton University. Email: pkrusell@princeton.edu.
†
1
1
Introduction
Consider an economy in which consumers all have the same preferences and income-generating
abilities but differ in wealth. Suppose, moreover, that there is collective decision making
over taxation and that the politically pivotal group is interested in redistributing resources
to itself. If individual-specific taxes or wealth-based taxes are available, such redistribution
is easy to accomplish. However, suppose that these redistribution tools are not available and
that taxes can only be levied on certain consumption goods; for simplicity, suppose that there
are two goods and that one of these goods can be taxed at a proportional rate, whereas the
other good cannot be taxed at all. If the proceeds from taxation are rebated in equal percapita amounts, then there is generally net redistribution from taxation, because rich and
poor consumers generally spend different amounts on the good that is taxed. In particular,
under a normal-goods assumption, a poor consumer would like a tax on the taxable good,
since rich consumers consume more of it and thus pay more in taxes than do poor consumers.
Put differently, poor agents would indeed set positive taxes to achieve redistribution, even if
taxation is distortionary: a small departure from zero taxes generates only a second-order loss
from distortions but a first-order gain from net redistribution.
In the present paper we apply this logic to two specific goods: a typical consumption good,
on the one hand, and leisure on the other. We take labor income to be taxable—i.e., we allow
leisure to be subsidized—whereas consumption taxes are not allowed. We thus examine the
idea that economies where taxation decisions are made by poorer-than-average consumers will
feature taxation of leisure: labor subsidies. We formalize the idea using an infinite-horizon
dynasty model where wealth differences arise from differences in initial wealth and preferences
admit aggregation.
Production uses labor only and the production function is linear in labor; our economy has
no physical or other form of real capital accumulation. Because the economy is also closed, this
means that in equilibrium average assets must be zero at each point in time; the real interest
rate adjusts so that this is the case. The dynamic economy differs from a static one primarily
because the interest rate is endogenous. Because of its impact on the total resources available
to consume, taxation influences interest rates. In particular, changes in the interest rate are
not neutral from the point of view of redistribution: a low interest rate, for example, is good
for debtors because it makes it cheaper to roll over loans, whereas for symmetric reasons, a
high interest rate is good for creditors. Analysis of the dynamic economy requires a careful
examination of how changes in taxes benefit different groups both through direct redistribution
and changes in interest rates, aside from the distortions of the labor-leisure choice they lead
to.
The possibility of “manipulation” of interest rates also leads to time-consistency problems:
politico-economic outcomes differ depending on whether or not there is a commitment device
for future policy. The reason is that a tax change today changes aggregate consumption and
therefore the interest rate between yesterday and today. Thus, ex post this effect will be
ignored if the tax rate is chosen now, but it would have been taken into account had the tax
choice been made in advance. The time-inconsistency problem means that in this economy,
there must be a difference in outcomes between two voting setups: one with commitment,
which we call time-zero voting, and one without, which we refer to as sequential voting.
Our findings are quite striking. In short, we point to (i) rather surprising features of the
commitment solution, and (ii) a stark qualitative difference between the solutions with and
without commitment. In the case of commitment, there is initial subsidization of labor—unless
the median voter is richer than average, in which case there is taxation—and asset inequal-
2
ity shrinks (median wealth, which is below zero to start with, moves toward zero). However,
after one period, taxes go to zero and stay there forever, and the asset distribution remains
unchanged. This finding, which is reminiscent of the results in Chamley (1986) and Judd
(1985)—that optimal capital taxes should be zero in the long run—is stark but it is not a
corollary of the Chamley/Judd characterization. It is not entirely general, because it uses the
assumption that preferences are in a certain class, however, the choice of preferences is motivated by the fact that they are used in almost all of the applied (quantitative) macroeconomic
literature.
We analyze the case without commitment using a Markov-perfect equilibrium notion, as
in the politico-economic equilibria studied in Krusell and Rı́os-Rull (1999).1 That is, politicoeconomic equilibrium taxes are a function of the payoff-relevant state and nothing else. We
use median-voter theory and obtain that the state variable in the no-commitment economy
is median wealth only. Here it is not easy to provide sufficient conditions for a median-voter
theorem. We thus verify single-peakedness numerically for the specific economies we look at.
We are not able to prove existence of a Markov-perfect equilibrium globally for our model but,
using the first-order condition, we can prove local existence around the point where median
assets equal mean assets.2 One can also see that, if an equilibrium exists globally and is smooth,
then it gives higher first-period subsidies than those found under commitment. Thus, a lack of
commitment leads to uniformly higher distortions: not only do subsidies remain positive from
period two and on, but they are also higher in the first period.
Mathematically, the functional first-order condition is different than in standard optimal
control theory problems since we are studying a dynamic game without commitment: the
median voter at a point in time cannot bind the hands of future median voters, whether the
latter are different individuals or not. Prior analysis focusing on first-order conditions in similar
contexts includes the work on individual saving under time-inconsistent preferences, where
reminiscent first-order conditions have been derived (so-called “generalized Euler equations”;
see Laibson [1997]), and some recent work on dynamic public finance (e.g., see Klein, Krusell,
and Rı́os-Rull [2005] or Azzimonti, de Francisco and Krusell [2006]). It turns out that the
numerical solution of this functional equation is challenging as well; we illustrate this fact in
the paper in a final computational section.
Motivation for our work can be found in the earlier political-economy literature. Meltzer
and Richard (1981) study a static model of the sort analyzed here but where heterogeneity
appears through labor productivity: some agents are more productive, and earn more, than
others. A dynamic version of their model without capital would allow asset accumulation on
the individual level, and it would allow interest-rate manipulation. However, this manipulation
would only have a role to play if agents had different initial asset positions—otherwise, the
dynamic Meltzer-Richard model would only be a mechanical repetition of the static one. Thus,
what we study here is the most stripped-down version of their economy with asset differences:
all the focus is on redistribution due to the wealth differences, and interest-rate manipulation is
central. It would seem straightforward to reintroduce productivity differences into the model,
which obviously would be a force toward taxation, not subsidization, of labor (assuming, as data
suggest, that median labor productivity is below mean labor productivity). The present work
also relates to Krusell and Rı́os-Rull (1999), who study a dynamic Meltzer-Richard model with
capital. In that model, there is a proportional tax on both labor and capital, and the focus is
mainly quantitative. In addition, the present paper focuses more on analytical characterization;
1
2
See Maskin and Tirole (2001) for a general discussion of the equilibrium concept.
From here on, our results apply to the case of logarithmic utility.
3
politico-economic equilibria in dynamic models are difficult to characterize, and numerical and
analytical work are both important. Furthermore, in any models where interest rates are
endogenous and consumers have different asset positions—features that ought to be present
in most quantitative political economy models—there is interest-rate manipulation, and the
politico-economic aspects of this manipulation therefore should seem to be of general relevance.
Finally, our median-voter result under commitment is related to Bassetto and Benhabib (2006),
who study a capital taxation economy with a similar preference structure; they use similar
reasoning to prove their median-voter result.
The paper is organized as follows. Section 2 describes the basic economic model for exogenous policy. This section demonstrates aggregation and characterizes equilibrium prices
(interest rates) and quantities as a function of the sequence of taxes chosen by the government.
Section 3 covers the political-economy model where there is commitment to future taxes. In
particular, that section demonstrates a median-voter theorem and characterizes the medianvoter equilibrium. Section 4 then defines and studies Markov-perfect equilibria when there is
no commitment. Section 5 concludes the paper.
2
The economic model
In this section, we describe the economic model for exogenous tax policy: we define sequential
competitive equilibria for given sequences of tax rates. We then specialize preferences to admit
aggregation and characterize equilibria in this case. This analysis will form the building block
for the later analysis of the politically determined policy.
2.1
Labor-leisure choice and wealth heterogeneity
In this infinite horizon economy, time is discrete and there is no uncertainty. Agents value
consumption, c, and leisure, l, and preferences are the same for everyone:
∞
X
β t u(ct , lt ).
t=0
Agents can trade one period bonds, each promising one unit of consumption next period.
The heterogeneity within the population enters only through differences in consumers’ initial
asset holdings. We will assume that the number of “types”
PI is finite with measure µi for type
i ∈ {1, 2, . . . , I}. Population size is normalized to one: i=1 µi = 1.
Production takes place according to a stationary production function which depends linearly on labor: Y = N (we use capital letters to denote aggregates). Output is used for
consumption, and there is no government consumption or investment. We also abstract from
capital entirely. The level of asset holdings of individual i is denoted by ai , which is in zero
net supply in equilibrium. Thus, neither storage nor trade with other countries is possible.
There is a constraint on the amount of time for each agent: each consumer has one unit,
so that li + ni = 1 for all i. We will make assumptions on primitives so that agents’ decision
problems are strictly concave; hence, all agents of the same type will make the same decisions
and we can also write Li + Ni = 1, where Li and Ni reflect common decisions
PI regarding leisure
and labor of all agents of type i. The aggregate labor input is thus N = i=1 µi Ni .
4
2.2
Sequential competitive equilibrium for given tax rates
In a decentralized economy, consumers buy consumption goods at each point in time and
sell their labor services to firms under perfect competition. There are two relevant prices
determined in equilibrium: the price of one-period bonds, denoted by q, and the wage rate
w, both measured in terms of consumption goods in the same period. In addition, we now
consider a government that taxes labor income at a proportional rate τt in period t and makes
equal lump-sum transfers Tt back to all consumers under a balanced budget. Thus, a typical
consumer i’s budget constraint in period t reads
cti + qt ait+1 = ait + wt (1 − lit )(1 − τt ) + Tt .
PI
In equilibrium, consumers’ holdings of assets have to add up to zero:
i=1 µi Ait = 0, where
Ait denotes the total holdings of agents of type i. Consumer heterogeneity thus originates in
ai0 6= aj0 for all i 6= j. We define a competitive equilibrium for a given sequence of government
policy as follows:
Definition 1 Given a tax policy {τt }∞
t=0 , a competitive equilibrium is a sequence of prices
{wt , qt }∞
together
with
a
sequence
of
allocations {Nt , {Cit , Ait+1 , Lit }Ii=1 , Tt }∞
t=0
t=0 satisfying the
following conditions.
1. For all i, {Cit , Ait+1 , Lit }∞
t=0 solves
∞
X
max
{ct ,at+1 ,lt }∞
t=0
β t u(ct , lt ).
t=0
subject to
ct + qt at+1 = at + wt (1 − lt )(1 − τt ) + Tt .
for all t, with a0 = Ai0 . Agents are also subject to a no-Ponzi scheme condition:
lim at Πt−1
k=0 qk = 0.
t→∞
2. For all t, wt = 1.
P
3. For all t, Nt = Ii=1 µi (1 − Lit ).
P
4. For all t, Ii=1 µi Ait = 0.
5. For all t, Tt = τt Nt .
An agent with wealth Ai0 will choose to accumulate assets and consume leisure so that
uc (cit , 1 − nit )wt (1 − τt ) = ul (cit , 1 − nit )
and
qt uc (cit , 1 − nit ) = βuc (cit+1 , 1 − nit+1 ).
Together with the budget constraints and a transversality condition, these conditions are sufficient for finding optimal consumer choices. In order to find an equilibrium, we need to
aggregate across consumers and find equilibrium bond prices in order to clear asset markets.
To accomplish this, we will make additional assumptions on preferences.
5
2.3
Aggregation
Our additional assumption on utility functions are made in order to deliver aggregation.
Assumption 1: Let the instantaneous utility be
u(ct , lt ) =
(cαt lt1−α )1−σ
.
1−σ
Under this assumption, the first-order conditions take a particularly simple form. As a
result, the competitive equilibrium aggregates and prices are independent of the distribution
of asset holdings.3
From agent i’s leisure choice, we see that all agents will choose the same ratio of consumption to leisure:
lit
1−α
=
.
cit
α(1 − τt )
(1)
The same agent chooses asset holdings to satisfy
1−α 1−σ
1−α 1−σ
qt (cαit lit
)
cit+1 = β(cαit+1 lit+1
)
cit .
By replacing equation (1) in the condition above we obtain
·
¸
cit+1
β 1/σ
=
.
cit
qt ²t
(2)
h
i(1−α)(1−σ)
t+1
. Thus, the growth rate of consumption is independent of idiowhere ²t = 1−τ
1−τt
syncratic characteristics. Finally, individual asset holdings must evolve according to
ait+1 = (ait + (1 − lit )(1 − τit ) + Tt − cit )/qt .
(3)
For completeness, we state
Proposition 1: Under Assumption 1 the competitive equilibrium exhibits aggregation.
Though the result is well known, we demonstrate it by a guess-and-verify strategy, since
the guess plays a role in the further characterization of equilibrium. By our guess, aggregate
quantities and prices can be found by solving the maximization problem of a representative
agent holding average wealth. WePwill thus refer to him as the mean agent. In other words,
his asset holdings are zero, since Ii=1 µi Ait = 0. The mean agent would consume goods and
leisure according to
1−α
Lt =
1 − ατt
and
Ct + (1 − τt )Lt = 1 − τt + Tt ≡ ft .
Here, we have written resources as the total net-of-tax value of time plus the transfer, which
we denote ft ; on the expenditure side, we have expenditures on consumption and leisure.
3
For a discussion of this result in an economy with capital see, e.g., Krusell and Rı́os-Rull (1999) or
Azzimonti, de Francisco and Krusell (2006).
6
In equilibrium Tt = τt (1 − Lt ). Therefore,
Ct = αf (τt ),
1−τt
where f (τt ) = 1−ατ
= ft . That is, the function f describes the economically relevant current
t
income of any agent in the economy—since it is based on the total amount of time, not hours
actually worked, and all agents have the same time endowment and productivity—as a function
of the current tax rate.
The first thing to note is that the consumption and leisure allocations of the mean agent
depend only on current taxes; this result really follows since assets of the representative agent
must be zero at all points in time (due to aggregation), and thus the representative-agent
economy reduces to a static one. So if the tax rates were constant over time, aggregates and
prices would also be constant. However, if they are not, there will be changes in the asset
distribution over time, as we will see below, even though mean assets remain at zero.
The first-order condition of an agent with average wealth implies
·
¸
Ct+1
β 1/σ
=
,
Ct
qt ²t
so the growth rate of consumption is identical to that chosen by an arbitrary agent in this
economy. It is immediate, now, that a summing up of leisure, consumption, and asset choices
of individual agents, using equations (1)–(2), delivers the expressions for averages given above.
Using the definition of consumption of the mean agent, we conclude that the price of bonds in
equilibrium must be
¶
µ
f (τt ) σ 1
.
qt = β
f (τt+1 )
²t
Thus, the interest equals the discount rate if and only if taxes are constant over time.
Aggregation allows us to characterize a competitive equilibrium by stating agents’ consumption and leisure decisions as proportional to mean consumption and mean leisure. The
proportion is given by their share of after-tax (potential) lifetime income in present value terms.
We now proceed to provide the appropriate expressions.
We can insert the equilibrium price of bonds, qt , into equation (2). The resulting growth
rate of consumption depends only on current and next period’s taxes:
cit+1
f (τt+1 )
=
,
cit
f (τt )
(4)
with
cit = α[f (τt ) + ait − qt ait+1 ];
under the present utility specification, consumption always commands a fraction α of total
resources left to spend this period, which are given by the right-hand side of this equation.
Similarly, leisure follows
lit =
1−α
[f (τt ) + ait − qt ait+1 ],
1 − τt
The present-value budget constraint of an agent with initial asset holdings ai0 is given by
∞
∞
X
1X
t−1
cit Πk=0 qk + ci0 = f (τ0 ) + ai0 +
Πt−1
k=0 qk f (τt )
α
t=1
t=1
|
{z
} |
{z
}
PV consumption
PV income
7
We can further simplify this expression and, using equation (4), solve for initial consumption
·
¸
ai0
ci0 = αf (τ0 ) 1 +
,
E(τ )
where the tax rate without subscript denotes the entire tax sequence, i.e., τ ≡ {τt }∞
t=0 , and
E(τ ) = f (τ0 ) +
∞
X
Πt−1
k=0 qk f (τt )
(5)
t=1
is the present value of net-of-tax time endowments and transfers received from the government.
It is also the lifetime income of the mean agent in present value terms. An arbitrary agent
is thus consuming a proportion of mean consumption (recall that the mean agent consumes
C0 = αf (τ0 )). This proportion is given by the share of the given consumer’s present-value
wealth, including transfers, the net-of-tax time endowment, and initial assets. It is given by
λi (τ ) ≡
E(τ ) + ai0
ai0
=1+
.
E(τ )
E(τ )
After some manipulation of equation (2), we thus find that
cit = αf (τt )λi (τ ).
| {z }
Ct
That is, the level of consumption of an agent with initial wealth ai0 , at any point in time, is a
proportion of average consumption.
Our equilibrium exists if λi (τ ) ≥ 0 for all i, i.e., if all agents can afford positive consumption
and leisure, which depends on the given tax sequence. This can be viewed as a restriction on
inequality, given a tax sequence, or as a restriction on tax sequences, given an asset distribution.
For some asset distributions, however, no tax sequences admit positive consumption, because
even the use of infinite subsidies to labor income is not enough to generate positive consumption
for the poorest. However, such initial asset distributions do not make sense, since they could
never have been chosen in the past.
Summarizing the information above, we can characterize the competitive equilibrium in
terms of the allocations chosen by the mean agent. The following proposition summarizes our
result.
Proposition 2: Under Assumption 1 and a given sequence of taxes τ = {τt }∞
t=0 satisfying
λi (τ ) ≥ 0 for all i, there exists a competitive equilibrium, and it is given as follows.
1. Transfers are Tt = ατt f (τt ).
2. The bond price is
µ
qt = β
f (τt )
f (τt+1 )
¶σ ·
1 − τt
1 − τt+1
¸(1−α)(1−σ)
3. Consumption and leisure allocations satisfy
cit = Ct λi (τ ) and lit = Lt λi (τ ) where
·
¸
ai0
λi (τ ) = 1 +
,
E(τ )
8
.
with E(τ ), the present value of net-of-tax time endowments plus transfers, is given by
equation (5). Per-capita values are
Ct = αf (τt )
and
Lt =
1−α
.
1 − ατt
4. Assets follow
ait+1 = (ait − [λi (τ ) − 1]f (τt ))
3
1
.
qt
Political economy: time-zero voting with commitment
The equilibrium characterization above implies that any agent who is poorer than average (i.e.,
with ai0 < 0) will enjoy less consumption and leisure than the mean agent: λi (τ ) < 1. While
this share is constant for a given stream of taxes τ , alternative tax sequences will modify it. It
is now clear where the core of the disagreement regarding policy in this economy lies: agents
realize that they can modify their share of average consumption and leisure by voting for the
appropriate τ . There is a cost associated to this manipulation, and it is given by the distortions
introduced in the labor decision. That is, alternative policies have different implications for
Ct or Lt , and we know that zero taxes is the best choice for the mean agent, thus leaving
Ct = αf (0) = α and Lt = 1 − α as his undistorted, best outcome. If an agent with non-zero
initial assets could decide, however, it is clear that a non-zero tax sequence would be chosen,
since there is a first-order marginal benefit from departing from zero taxes—the change in the
share of resources—and only a second-order loss.
We now need to be specific about the political institution. In this section, we will assume
that voting takes place at the beginning of time, and that the policy-setting power has a
commitment technology at that point: it can fix a sequence of taxes for all time. We assume
that political competition takes place in the form of direct votes over tax sequences, and we
assume majority voting.
Definition 2: A majority-voting equilibrium (MVE) is a tax sequence that cannot
be beaten in a pairwise voting contest by any other sequence.
A tax sequence is a high-dimensional object and it is thus far from obvious that this
definition has content in our economy—it is well known that the existence of an MVE cannot
be guaranteed except under special circumstances. Here, fortunately, the same assumption
on preferences that admits aggregation will allow us to show existence of a majority-voting
equilibrium: we will provide a median-voter theorem.
We need to derive indirect utility expressions for the different agents, first, which are
available in closed form under our assumptions. It is easy to verify that they are given by
Vi0 (τ ) = λi (τ )1−σ V0 (τ ),
where V0 (τ ) is the utility of the agent with average wealth, given by
·
¸1−σ
∞
X
f (τt )
t
β
V0 (τ ) = η
,
(1 − τt )1−α
t=0
9
(6)
£ α
¤1−σ
1
with η = 1−σ
α (1 − α)1−α
. What is important to note here, from the purposes of
existence, is that λi (τ ) is linear in the only element of heterogeneity in the model: in ai0 .
Let τ ∗ be our candidate majority-voting equilibrium, then it must be the case that this
policy is preferred by at least 50% of the population to any other policy. We can show that
the equilibrium tax sequence is that preferred by the agent with median wealth, the median
agent.
3.1
The median-voter theorem
We denote the wealth of the median agent by a∗ . Similarly, his preferred tax sequence is
denoted τ ∗ . Thus,
µ
¶1−σ
µ
¶
a∗
a∗ 1−σ
∗
1+
V0 (τ ) ≥ 1 +
V0 (τ )
(7)
E(τ ∗ )
E(τ )
for all feasible τ , i.e., for all τ such that λi (τ ) ≥ 0 for all i.
We now state our median-voter theorem.
Proposition 3: τ ∗ is a majority-voting equilibrium.
The proof is straightforward, but perhaps instructive. For any alternative feasible τ , we
know that
¶1−σ
¶
µ
µ
a∗
a∗ 1−σ
∗
1+
V0 (τ ) ≥ 1 +
V0 (τ ).
E(τ ∗ )
E(τ )
This implies, using simple manipulations, that either
³
a∗ ≥
1
E(τ ∗ )
or
−
³
a∗ ≤
1
E(τ ∗ )
V0 (τ )
V0 (τ ∗ )
´
1
E(τ )
1
1−σ
³
1
E(τ )
V0 (τ )
V0 (τ ∗ )
−
´
V0 (τ )
V0 (τ ∗ )
1
1−σ
³
−1
´
−1
´
V0 (τ )
V0 (τ ∗ )
1
1−σ
1
1−σ
,
depending on the signs of the numerators and denominators in these expressions. Therefore,
since half of the population has ai0 ≥ a∗ and the other half, ai0 ≤ a∗ , working in reverse, it
must be that for half of the population, there is agreement with the choice of the median agent:
for more than half the population, their ai0 is such that
¶1−σ
µ
¶
µ
ai0 1−σ
ai0
∗
V0 (τ ) ≥ 1 +
V0 (τ ).
1+
E(τ ∗ )
E(τ )
3.2
Characterization
We now proceed to show that, under the preference specification adopted, subsidies are zero
from period 1 and on, whereas they are negative in period 0 if and only if a∗ < 0.
The key step in the proof is the demonstration that, under our preference specification,
E(τ ) can be written as a product of a function only of period-0 taxes, g(τ0 ), and the presentvalue utility of the mean agent:
10
Lemma: E(τ ) = g(τ0 )V0 (τ ), with g(τ0 ) ≡ η −1 f (τ0 )σ (1 − τ0 )(1−α)(1−σ) .
The proof is contained in Appendix 6.1.
Thus, we conclude that the utility of the median agent can be written as
·
a∗
V∗0 (τ ) = 1 +
g(τ0 )V0 (τ )
¸1−σ
V0 (τ ).
(8)
Now we verify the following.
Proposition 4: Assume that σ ≤ 1. The politico-economic equilibrium then has the
following properties, assuming a∗ < 0:
1. For all t > 0, τt = 0.
2. τ0 < 0.
3. q0 > β and qt = β for all t > 0.
4. For all i such that ai0 < 0, ai0 < ai1 = ai2 = · · · < 0 and for all i such that ai0 > 0,
ai0 > ai1 = ai2 = · · · > 0.
The proof of the first two parts of the proposition is contained in Appendix 6.2. In brief, the
proof of the first part involves a straightforward implication of the lemma above, namely, that
the first-order condition with respect to τt is zero at τt = 0, since τt for t > 0 only appears
through V0 (τ ), which we know is maximized at zero taxes; it also involves the verification of
a second-order condition. The second-order condition necessitates restricting −σ from being
too large relative to the median voter’s initial debt position; values of σ above one can thus be
admitted as well, so long as a∗ is above a certain lower bound. The proof of the second part
is straightforward. The third and fourth parts of the proposition are immediate implications
of the first two parts, together with the earlier characterization of equilibrium prices and
quantities for given tax sequences.
The result that taxes are zero in the long run (and even from the second period and on)
is surprising; one might think that the redistribution generated by subsidies to labor should
not just take place at time zero—that at least a minor amount of redistribution in the future
would be better than none. The proposition proves this intuition wrong. Zero taxes in the
long run is also reminiscent of the findings of Chamley (1986) and Judd (1985). Here, though
it is not an optimal taxation problem, the median agent does have an accumulation problem,
and the tax sequence influences the return on savings as well as a period income function f (τt )
similar to the Chamley-Judd settings. We do find that savings are undistorted in the long run,
but this outcome would result for any constant tax on labor income, whether positive, zero, or
negative. The fact that we find that the labor income tax will be exactly zero is, thus, what is
different, and surprising. Therefore, there is really no obvious connection between the result
here and the Chamley-Judd finding, since zero taxes has a different meaning in the two cases.
The preference specification is key behind the result. In fact, it is possible to extend our
result to a utility function that is a power function of a CES function in consumption and
leisure (and not just a unitary-elasticity Cobb-Douglas function). However, if the elasticity
of substitution is not constant between consumption and leisure, one can demonstrate that
the long-run tax is non-zero. Thus, it is the fact that, under homothetic utility specifications,
11
wealth differences do not misalign preferences over tax rates enough that is driving our main
finding. There are four kinds of effects when a change of τt away from zero is considered. The
first two involve the effect on redistribution and the associated distortion to the consumptionleisure choice. That effect would, in isolation, lead the median to choose a negative tax
rate. The two other effects involve interest-rate manipulation. A decrease in τt will increase
labor supply, which will decrease marginal utility of consumption at t, thus raising qt but
lowering qt−1 . The former effect is beneficial for the median voter, but the latter is detrimental.
Evaluated at τt = 0 for all t > 0, the net marginal effect of these two changes is negative, and
it exactly outweighs the redistributional benefits of lowering the tax rate.4
4
Sequential voting without commitment
We now move to the context where there is no commitment technology and taxes for period
t are voted on in the beginning of period t (before work decisions have been made). For this
environment, we employ the notion of Markov-perfect equilibrium, as used in Krusell and
Rı́os-Rull (1999). The set of Markov-perfect equilibria is likely large (see the discussion in
Krusell and Smith [2003]), and we restrict attention here to Markov-perfect equilibria that
would appear as limits of equilibria in the corresponding finite-horizon economies. Thus, we
naturally focus on equilibria where equilibrium taxes are a smooth function of the payoffrelevant state variable.
The payoff-relevant state variable in our economy is whatever feature of the asset distribution matters for the political outcome, and no other feature. Again, given taxes, and due
to Assumption 1, there is aggregation, and nothing else than mean asset holdings can matter
from the economic perspective: in the last period, no other feature matters for outcomes, and
thus in the period before the last, no other feature matters either, and so on.5 What are
the politically relevant features of the asset distribution? We will conjecture that there is a
median-voter characterization here as well, and we will thus proceed based on this conjecture.
In conclusion, the state variable is a∗ : the asset holdings of the median agent. From this point
on, for simpler notation, we will merely use a to denote median asset holdings.
4.1
Equilibrium definition based on a simple conjecture
We will not define a Markov-perfect equilibrium in full generality, since it consumes space, and
such a definition would closely follow the definitions in the literature. Instead we will state our
conjecture and define an equilibrium based on it.
We conjecture that taxes in any period are a function of a: τ = Ψ(a). In other words,
Ψ is the fundamental endogenous object we are in search of. Given this conjecture, it should
be clear that taxes, and the asset distribution, will be constant from the beginning of time.
4
In fact, the result that future taxes are zero holds also in finite-horizon settings, perhaps surprisingly.
There, one might think that the taxation decision in the last period would lead to a different outcome,
since only one interest rate is affected by the last period’s tax rate. There, however, the effect of
increasing the second-to-last period’s q again exactly offsets the gain from redistribution due to a lastperiod tax cut.
5
It is conceivable that one can construct a Markov-perfect equilibrium in a political-economy model
where other features of the asset distribution matter, even when preferences admit aggregation, but
then as a “multiple-equilibrium” phenomenon: other features matter now because they matter in the
future. Such equilibria will, however, not be limits of finite-horizon equilibria.
12
Why should this be true? Suppose that, given any function Ψ, we (rationally) expect taxes
not to change. Then, q will equal β, and the asset distribution—in particular, median asset
holdings—will not change. As a result, taxes next period will indeed be the same, since they
are given by the function Ψ, evaluated at the median asset level. Thus, the conjecture of
a constant tax rate is internally consistent. However, could different expectations regarding
future taxes be self-fulfilling? Assuming that Ψ is differentiable, one can show that the answer
is negative, at least in the neighborhood of a = 0.6
Proceeding with the guess, we compute the median consumer’s utility of any next-period
asset level as follows:
¡
¢1−σ
c(a)α l(a)1−α
V (a) ≡
,
(1 − β)(1 − σ)
where
µ
c(a) ≡ 1 +
and
µ
l(a) ≡ 1 +
a
e(Ψ(a))
a
e(Ψ(a))
¶
αf (Ψ(a))
¶
(1 − α)
f (Ψ(a))
,
1 − Ψ(a)
with the function e defining the present-value wealth given the constant tax sequence:
e(Ψ(a)) ≡
1
f (Ψ(a)).
1−β
We now consider a one-period deviation from the constant-tax equilibrium, i.e., that taxes
in the current period are not necessarily given by τ = Ψ(a). Now it is useful to define present
consumption and leisure
¶
µ
a
αf (τ )
(9)
c̃(a, a0 , τ ) ≡ 1 +
ẽ(a, a0 , τ )
and
˜l(a, a0 , τ ) ≡
µ
1+
a
ẽ(a, a0 , τ )
¶
(1 − α)
f (τ )
,
1−τ
(10)
with
ẽ(a, a0 , τ ) ≡ f (τ ) + q̃(a0 , τ )e(Ψ(a0 ))
and
µ
0
q̃(a , τ ) = β
f (τ )
f (Ψ(a0 ))
¶σ ·
1−τ
1 − Ψ(a0 )
(11)
¸(1−α)(1−σ)
,
the current bond price.
Thus, we can define our Markov-perfect equilibrium as follows:
Definition: A Markov-perfect equilibrium is a function Ψ solving
³
Ψ(a) = arg max
τ
c̃(a, a0 , τ )α ˜l(a, a0 , τ )1−α
1−σ
6
´1−σ
+ βV (a0 ),
To do this, one can (i) express asset accumulation of the median agent as a function of current
a, current taxes Ψ(a), and future taxes Ψ(a0 ), where a0 is next period’s median asset holdings, (ii)
differentiate the expression, and (iii) evaluate at a = 0. This delivers a unique linear solution: a0 = a.
13
where a0 is given implicitly by the solution to
·
¸
1
1
0
f (τ )
.
a =a 1−
0
ẽ(a, a , τ )
q̃(a0 , τ )
The latter constraint is the asset accumulation equation, slightly simplified, from Proposition
2.
The median-voter theorem holds if the objective above is single-peaked in τ for every a.
The shape of the objective is, of course, influenced by Ψ, which itself is endogenous; however,
single-peakedness can be verified numerically.
4.2
Analytical findings and comparisons with the commitment
solution
In this section, we partially characterize the Markov-perfect equilibrium and compare it to
the commitment case. In the next section, we continue our analysis using numerical solution
techniques.
From now on, we specialize to the case of σ = 1 (logarithmic utility). Under this assumption, the present-value wealth from time endowments, net of taxes, and transfers, under a
one-period deviation, is independent of asset holdings. Thus equation (11) reduces to
ẽ(τ ) =
f (τ )
,
1−β
while the price of bonds becomes
q̃(a0 , τ ) = β
f (τ )
.
f (Ψ(a0 ))
Consequently, a0 is implicitly defined by
a0 = a
f (Ψ(a0 ))
.
f (τ )
(12)
Equation (12) determines the saving rule for the median voter, which we denote by h̃(a, τ ).
By replacing the equations above into equations (9) and (10) we obtain expressions for consumption and leisure in the logarithmic case:
£
¤
c̃(a, a0 , τ ) = α f (τ ) + a − q̃(a0 , τ )a0
and
0
˜l(a, a0 , τ ) = (1 − α) c̃(a, a , τ ) .
α
1−τ
0
0
Notice that q̃(a , τ )a = βa, which implies that we can write
c̃(a, τ ) = α [f (τ ) + (1 − β)a] .
The Markov-perfect equilibrium tax function Ψ(a) then solves
max
0
a ,τ
α log c̃(a, τ ) + (1 − α) log ˜l(a, τ ) + βV (a0 )
14
where
V (x) ≡
ª
1 ©
log α2α−1 (1 − α)1−α + log [f (Ψ(x)) + x(1 − β)] − (1 − α) log[1 − Ψ(x)] ,
1−β
subject to equation (12), i.e., subject to a0 = h̃(a, τ ).7 It is clear here how the choice for
current taxes also influences future outcomes through the effect on future taxes in this model.
In particular, τ 0 = Ψ(a0 ) = Ψ(h̃(a, τ )).
Using the assumption that Ψ is differentiable, we can derive a first-order necessary condition. It reads
½
1
1
1
a
−
+β
(13)
1−τ
1 − ατ 1 − τ + a(1 − β)(1 − ατ )
(1 − τ )[1 − τ + a(1 − β)(1 − ατ )]
¾
αΨ(a0 )
0
−
Ψa (a )h̃τ (a, τ ) = 0,
(1 − β)(1 − Ψ(a0 ))(1 − αΨ(a0 ))
where h̃τ (a, τ ) is given by implicit differentiation of equation (12).8
The first two terms in equation (13) summarize the static trade-off between the net gain
from redistribution resulting from a positive subsidy with the negative effect of the introduced
distortion in the leisure-labor decision; it is negative at τ = 0 if and only if the median agent
is poorer than the median, a < 0. This would be the complete analysis in a Meltzer-Richard
(static) version of our economy. The remaining part of the first-order condition—the second
and third terms, which are both of second order at a = 0—capture how future utility is affected
by a change in current taxes. The second term incorporates the effects of an induced change
in current prices (q̃(a0 , τ )) on future consumption and leisure via changes in bond holdings,
assuming that the tax rate next period remains unaffected (that is, keeping Ψ(a0 ) constant).
The last term describes the indirect effect on utility through how the current tax rate alters
future tax rates.
In a differentiable Markov-perfect equilibrium, condition (13) needs to deliver τ = Ψ(a):
thus Ψ constitutes a fixed point. Moreover, a0 will equal a in the equilibrium construction.
Therefore, we can state a functional equation in Ψ by imposing these conditions in the equations
above, resulting—after some simplifications—in the following:
1−αΨ(a) =
αβΨ(a)(1 − Ψ(a))(1 − αΨ(a))
1 − Ψ(a)
−
. (14)
1 − Ψ(a) + a(1 − β)(1 − αΨ(a)) (1 − Ψ(a))(1 − αΨ(a)) + a(1 − α)Ψa (a)
This functional equation determines Ψ as a function of a: Ψ has to be such that the equality
is met for all values of a. The equation can be labeled a “generalized Euler equation”, a term
also used in other settings with dynamic decisions made without commitment and a Markovperfect equilibrium concept, such as the case with savings under time-inconsistent preferences
7
Here, we use the fact that since consumption can be written as a function of τ and a only, ˜l(a, τ ) =
(1−α) c̃(a,τ )
α
1−τ .
8
The derivation of equation (13) involves an intermediate step where we substitute equation (12),
where x = a f (Ψ(x))
f (τ ) , and use the properties of logarithmic functions to separate the direct effects of τ
on V (x) from those resulting from indirect changes through Ψ(x). Using a0 as shorthand for h̃(a, τ ),
implicit differentiation of (12) delivers
h̃τ (a, τ ) =
a0 (1 − αΨ(a0 )) − aα(1 − Ψ(a0 ))
.
(1 − τ )(1 − αΨ(a0 )) − (αa0 (1 − τ ) − a(1 − ατ ))Ψa (a0 )
15
(see, e.g., Laibson, 1997). In these cases, the first order condition also contains the derivative
of the unknown policy function. We will discuss the characterization of Ψ(a) in the context of
comparing it to the commitment solution.
Under time-zero voting, we can directly apply the insights from Section 3. However, it is
useful to take one step back and restate the commitment problem here, because this allows for
a convenient parallel to the problem without commitment. The problem can be stated
max
{τt ,at+1 }∞
t=0
∞
X
t=0
·
β
t
¶
¸
µ
f (τt )
log f (τt ) + at − β
at+1 − (1 − α) log(1 − τt ) .
f (τt+1 )
This problem is unconstrained: one can easily show that the choice for asset holdings delivers
at+1 = at
f (τt+1 )
,
f (τt )
which would have been the constraint representing equilibrium choices (thus, maximization
over a larger choice set leads to a choice within the restricted set, so the restriction can be
ignored).
Knowing that taxes from period 1 and on are constant, and that this implies that bond
holdings are also constant from period 1 on, we obtain, for all a, that taxes in period zero
satisfy
1
1
a
1
−
+β
= 0, (15)
1 − τ0 1 − ατ0 1 − τ0 + a(1 − β)(1 − ατ )
(1 − τ0 )[1 − τ0 + a(1 − β)(1 − ατ0 )]
where a now refers to a0 , for ease of comparison with the no-commitment solution. That is,
τ0 denotes the first-period choice for taxes under commitment. Slightly abusing notation, we
denote the tax function that solves the equation above by τ0 (a).
Comparing equations (13) and (15) we see only one difference: the former—the no-commitment
case—has an extra term. The reason is that under a period-by-period voting mechanism, the
current policymaker does not have a direct control over future taxes: it can only affect them
indirectly via the level of a0 . If the policy function were constant, i.e., if it did not depend on
a, so that Ψa (a) = 0 in the expression above, then the commitment solution would be timeconsistent. However, we know that it cannot be, because the commitment solution does deliver
a tax rate that depends on a: simple inspection of equation (15) reveals that τ0 (a) cannot be
independent of a. Thus, in general, Ψ(a) 6= τ0 (a). We see, however, that for a = 0, so long as
Ψ(a) is differentiable at 0, both equations (13) and (15) deliver zero taxes: Ψ(0) = τ0 (0) = 0.
That is, if the median asset holdings equal the mean, then taxes are zero, no matter how voting
takes place. The median voter chooses not to distort labor decisions because he cannot redistribute resources in his favor. Since the median asset holdings do not change (he chooses to
hold no debt), taxes in the future will also be zero under commitment. There are no incentives
to deviate from this policy once period one arises, and thus the policy is time-consistent.
It is also possible to compare the cases with time-zero voting and sequential voting in the
neighborhood of equality of median and mean income (a = 0). First, note that initial taxes
under commitment allow for a closed-form solution, and that the implied slope of τ0 (a) can be
16
determined.9 Thus, it follows in particular that the slope of τ0 (a) at the origin is given by
¯
dτ0 ¯¯
1
= ,
da0 ¯a0 =0 α
i.e., that at least locally around the point where median and mean income coincide, initial taxes
are increasing at the rate 1/α > 1 in median asset holdings. Thus, the higher is the preference
weight on consumption relative to leisure (α), the less do subsidies respond to inequality
between median and mean income. Intuitively, a low relative value of leisure makes the labor
effort differ less across wealth groups, so for each a there is a smaller income discrepancy across
groups and, hence, lower benefits from redistribution.
Interestingly, around the point where median wealth equals mean wealth, the local (firstorder) sensitivity of first-period taxes to inequality is the same under sequential voting as
under time-zero voting, provided that Ψ is twice differentiable at zero. Showing this, i.e., that
Ψa (0) = 1/α, is straightforward: it amounts to differentiating equation (14) with respect to
a, then evaluating the resulting expression at a = 0, and verifying that Ψa = 1/α solves the
equation.10
Furthermore, it is possible to see that, in the neighborhood of a = 0 but for strictly negative
values of a, subsidies under the no-commitment case are higher than under commitment, i.e.,
that
Ψ(a) < τ0 (a).
This can be seen as follows. For a small negative level of debt, the third term of equation
(13) is negative since (i) the term in brackets is positive, because the median subsidizes when
0
a < 0; (ii) Ψa (a) > 0 at zero; and (iii) dτ
dτ = Ψa (a)h̃τ (a, τ ) < 0, which can be shown by using
the fact that Ψ(a) is increasing in this neighborhood. Given this, at the commitment tax level,
the first term is zero while the second one is negative; hence, the equality in equation (13)
cannot hold. The median must then adjust τ by increasing the first term. This can only be
done by decreasing τ , since the first term is decreasing in τ for small as. These differences will
be illustrated numerically in the next section. The reason why subsidies are higher in the first
period when there is no commitment comes from the fact that interest-rate manipulation is
actually easier when there is no commitment. To see why, from equation (12), a raised subsidy
today will make the median asset holding come closer to zero: first, it will lower a0 /a since it
raises consumption today directly and hence lowers the interest rate; second, since a0 /a tends
to fall, there is a feedback through the lower implied subsidy rate tomorrow, further lowering
the interest rate. Thus, the feedback effect of lower future subsidies coming from a raised
subsidy today, which is not present under commitment because future subsidies are always
zero in that case, increases the interest-rate effect of a given subsidy change when there is a
lack of commitment.
9
One obtains a second-order polynomial expression for τ0 for each a, yielding
p
−α[−1 + a0 (β − 2)] − α2 [−1 + a0 (β − 2)]2 − 4a0 α(1 + a0 α(1 − β))
∗
τ0 =
.
2α(1 + a0 α(1 − β))
10
The expression involves the product of the second derivative of the policy rule, Ψaa (a), and the
level of assets a. The term cancels out at a = 0 under the assumption that Ψaa (a) < ∞.
17
4.3
A numerical example and a computational challenge
We proceed using the case of logarithmic utility. Using the method described in Krusell,
Kuruşçu, and Smith (2002), we solve the model numerically using the parameter values α = 0.3
and β = 0.9.11 It delivers the following results in a neighborhood of a = 0. Figure 1 illustrates
Labor subsidies under commitment (Comm) vs
subsidies in the Markov−perfect equilibrium
0
−0.002
Tax rate
−0.004
−0.006
−0.008
MP
Comm
−0.01
−0.012
−3
−2.5
−2
−1.5
A
m
−1
−0.5
0
−3
x 10
Figure 1: Numerical solution of the model under logarithmic utility
the features derived in the previous section: the commitment and Markov-perfect solutions
that coincide at zero asset holdings of the median have the same slope at that point, with the
commitment solution giving lower subsidies for strictly negative values of median assets.
It turns out that the model here presents a formidable computational problem. Figure 1
displays the tax function in a very small neighborhood around zero median asset holdings. It is,
however, very difficult to extend the results beyond this small neighborhood, and we contend
that it is an open problem how to best approximate equilibria in cases like this one. As an
illustration, we include numerically computed first-period policy rules (median asset accumulation and the chosen tax rate) for a sequence of economies with an increasing time horizon:
we look at horizons of length 1 through 9. These decision rules are solved by discretizing the
space of asset holdings, using a very fine grid. Figures 2 and 3 report the results.
As can be seen, first, in Figure 2, assets move toward zero, since the model has a finite
horizon; for longer time horizons, the asset rule is closer to the 45-degree line, as expected (in
the infinite-horizon Markov-perfect equilibrium, assets are constant over time, i.e., the decision
rule is the 45-degree line).
In Figure 3 we see the decision rules for taxes. As expected, the tax function is positively
sloped, goes through zero, and is lower, the longer the time horizon: the future benefits of
redistribution engineered by raising q in the first period are higher, the longer the future
period over which the higher asset holdings are used. However, what is probably more eyecatching in the figure is the fact that wiggles appear in the tax function at longer horizons.
We have not been able to find ways of making sure that no wiggles appear. The numerical
problems may be related to the fact that one expects the equilibrium tax rate to approach −∞
11
We have also used a pure grid method, based on the definition of the equilibrium involving value
functions. It reproduces these results.
18
Asset holdings as we extend the time-horizon
0
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
-0.1
A'm
-0.2
-0.3
-0.4
A'_1
A'_2
A'_3
A'_4
A'_5
A_6
A'_7
A'_8
A'_9
45o
-0.5
-0.6
Am
Figure 2: Next-period assets in the first period for a sequence of finite-horizon economies
as median asset holdings approach its lowest feasible value. For the infinite-horizon model,
a
1
this value, a, solves 1 + E(−∞)
= 0; for logarithmic utility, a = − α(1−β)
. Numerical instability
of Markov-perfect equilibria has been noted before, beginning in Krusell and Smith (2003).
The idea there, loosely speaking, is that wiggles/numerical errors, once they appear, tend to
reproduce and amplify due to the time-inconsistency inherent in the decision problem. This
delivers a non-concave objective function and, eventually, discontinuous behavior.
5
Conclusions
With this paper, we continue the exploration of dynamic macroeconomic models of political
economy. We focus on a case where the only source of disagreement on policy is the heterogeneity of initial asset positions: initially well-endowed consumers choose to work less, and
a poor median voter thus wants to subsidize leisure. The case is interesting and nontrivial
in part because interest rates are endogenous and call for manipulation by the median voter:
they are adjusted so as to change his net-present-value wealth position, in effect, by raising
the interest rate so that the debt he carries is an easier burden.
The tax sequence chosen by the median voter under commitment involves zero taxes from
period 1 and on, presuming that preferences are in the class admitting aggregation. We ensure
that a median-voter equilibrium obtains under the same conditions. The tax choice is not
time-consistent, however: without commitment, taxes will be constant (and non-zero) over
19
Taxes as we extend the time-horizon
0
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
-0.5
<(Am)
-1
-1.5
-2
tau_1
tau_2
tau_3
tau_4
tau_5
tau_6
tau_7
tau_8
tau_9
-2.5
-3
Am
Figure 3: Taxes in the first period for a sequence of finite-horizon economies
time. Asset inequality changes between the first two periods in the case with commitment—
inequality is reduced—but under lack of commitment, there is no change in asset inequality,
despite the power of the median voter to subsidize labor income.
Open issues remain, particularly regarding the characterization of Markov-perfect equilibria. Even though we show that the present model, like the model of savings under quasigeometric preferences, reduces to a simple-to-express functional equation—equation (14), which
has one unknown function of one variable—it is surprisingly difficult to characterize equilibrium globally, even numerically. It is therefore an outstanding challenge to search for numerical
methods aiding the analysis of Markov-perfect equilibria in this and similar kinds of models.
20
6
Appendix
6.1
Proof of Lemma
We have that
E(τ ) = f (τ0 ) +
∞
X
f (τt )Πt−1
k=0 qk
(16)
t=1
and that
µ
qt = β
f (τt )
f (τt+1 )
¶σ ·
1 − τt
1 − τt+1
¸(1−α)(1−σ)
.
It follows that the discount factor Πt−1
k=0 qk equals
µ
β
f (τ0 )
f (τ1 )
¶σ ·
1 − τ0
1 − τ1
¸(1−α)(1−σ)
µ
×β
f (τ1 )
f (τ2 )
¶σ ·
1 − τ1
1 − τ2
¸(1−α)(1−σ)
µ
× ... × β
f (τt−1 )
f (τt )
¶σ ·
1 − τt−1
1 − τt
This can be simplified to
µ
Πt−1
k=0 qk
=β
t
f (τ0 )
f (τt )
¶σ ·
(1 − τ0 )
(1 − τt )
¸(1−α)(1−σ)
.
Inserting this expression back into equation(16), we obtain
E(τ ) = f (τ0 ) +
∞
X
·
t
σ
(1−α)(1−σ)
β f (τ0 ) (1 − τ0 )
t=1
f (τt )
(1 − τt )(1−α)
¸1−σ
,
which can be reduced to
E(τ ) = f (τ0 )σ (1 − τ0 )(1−α)(1−σ)
∞
X
·
βt
t=0
Since
∞
X
t=0
·
β
t
f (τt )
(1 − τt )(1−α)
¸1−σ
=
f (τt )
(1 − τt )(1−α)
¸1−σ
.
V0 (τ )
,
η
we can write the present value of net income and transfers as a function of the utility of the
representative agent, i.e.,
E(τ ) = g(τ0 )V0 (τ ),
where g(τ0 ) = η −1 f (τ0 )σ (1 − τ0 )(1−α)(1−σ) .
6.2
Proof of Proposition 4
As a first step it is useful to characterize the optimal tax sequence for the representative agent.
Taxes must satisfy
·
¸−σ
∂V0
f (τt )
(1 − α)ατt
t
=β η
= 0 ∀t.
(1−α)
∂τt
(1 − τt )1−α (1 − ατt )2
(1 − τt )
This agent will thus prefer τt = 0∀t to any other sequence. Intuitively, this agent obtains no
net redistribution (the taxes paid amount to the transfers received) and does not benefit from
21
¸(1−α)(1−σ)
.
changing interest rates, since she has no assets. There is no gain from taxing or subsidizing
labor, but a second-order loss generated by the distortion it generates on labor decisions. As
a result, she finds it optimal to set taxes equal to zero. The median agent on the other hand,
can gain from net redistributions and the manipulation of the interest rate sequence.
The indirect time-zero utility function of an agent with wealth ai0 can be written as a
function of the mean agent’s utility and the tax rate at time 0, as we have shown above. That
is,
·
¸1−σ
ai0
Vi0 (τ ) = 1 +
V0 (τ ).
g(τ0 )V0 (τ )
The first-order condition with respect to taxes at time t ≥ 1 is
·
¸
∂Vi0
(1 − σ)ai
∂V0 (τ )
1−σ
= λi (τ )
1−
= 0,
∂τt
g(τ0 )V0 (τ )λi (τ )
∂τt
so the median will choose τt = 0 for any t ≥ 1.
We need to make sure that this is indeed a maximum. The second order condition evaluated
at the optimum is
¸ 2
·
∂ 2 Vi0
(1 − σ)ai
∂ V0 (τ )
1−σ
=
λ
1
−
g(τ0 )V0 (τ )λi (τ )
∂τt2
∂τt2
2
Then ∂∂τV2i0 < 0 if and only if the term in brackets is positive. The latter will hold as long as
t
the agent’s initial asset holdings satisfy
ai0 > −
g(τ0 )V0 (τ )
.
σ
If σ ≤ 1, this condition is met. In order for the competitive equilibrium allocations to be well
defined we need the i’s share of income λi ≥ 0. But that translates exactly to the condition
ai0 > −g(τ0 )V (τ ). Thus, this condition is met whenever σ ≤ 1.
On the other hand, when σ > 1 the condition is not automatically satisfied. In this case,
the second-order conditions might be violated if ai0 is too low, which imposes a restriction on
the degree of initial inequality for which a zero tax policy will be voted in equilibrium.
22
References
[1] Azzimonti, M., de Francisco, E. and Krusell, P. (2006) Median-Voter Equilibria in the
Neoclassical Growth Model Under Aggregation. Manuscript.
[2] Bassetto, M. and Benhabib, J., 2006, Redistribution, Taxes and the Median Voter, Review
of Economic Dynamics, 9:2, 211–223.
[3] Chamley, C., 1986, Optimal Taxation of Capital Income in General Equilibrium with
Infinite Lives, Econometrica, 54, 607–22.
[4] Judd, K., 1985, Redistributive Taxation in a Simple Perfect Foresight Model, Journal of
Public Economics, 28, 59–83.
[5] Klein, P., Krusell, P. and Rı́os-Rull, J.V., 2005, Time-Consistent Public Expenditures,
Working Paper.
[6] Krusell, P., Kuruşçu B., and Smith, A., 2002, Equilibrium Welfare and Government Policy
with Quasi-Geometric Discounting, Journal of Economic Theory, 75, 42–72.
[7] Krusell, P. and Smith, A., 2003, Consumption-Savings Decisions with Quasi-Geometric
Discounting, Econometrica, 71, 365–375.
[8] Krusell, P. and Rı́os-Rull, J.V., 1999, On the Size of Government: Political Economy in
the Neoclassical Growth Model, American Economic Review, 89:5, 1156–1181.
[9] Laibson, D., 1997, Golden Eggs and Hyperbolic Discounting, Quarterly Journal of Economics, 112:2, 443-77.
[10] Maskin, E. and Tirole, J., 2001, Markov Perfect Equilibrium, Journal of Economic Theory,
100:2, 191–219.
[11] Meltzer, A.H. and Richard, S.F., 1981, A Rational Theory of the Size of Government,
Journal of Political Economy, 89:5, 914–27.
23
@FedFRASER