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DEWEY ll
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
DYNAMIC MIRRLEES TAXATION UNDER
POLITICAL ECONOMY CONSTRAINTS
Daron Acemoglu
Michael Golosov
Aleh Tsyvinski
Working Paper 08-08
February 15, 2008
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
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http://ssrn.com/abstract=1
HB31
.M415
at
Dynamic Mirrlees Taxation under
Political
Economy
Constraints*
Daron Acemoglu
Michael Golosov
MIT
MIT
Aleh Tsyvinski
Harvard and NES
February 2007.
Abstract
We
study the structure of nonlinear incentive-compatible taxes, in a dynamic economy subject
to political
economy and commitment problems.
In contrast to existing analyses of
dynamic and/or
we relax the assumptions that taxes are set by a benevolent government
commitment to policies. Instead, in our model economy taxes are set by a self-
nonlinear taxation problems,
and that there
is
interested pohtician, without any
via elections.
We
The
commitment power. This
resulting environment
is
Towards a
of the citizens.
mechanism, we
first
full
is
partly controlled by the citizens
one of a dynamic mechanism design without commitment.
focus on the best sustainable mechanisrn, which
utilitj'
politician
is
the
mechanism that maximizes the ex ante
characterization of the allocations implied by the best sustainable
prove that a version of the revelation principle applies in our environment and
that attention can be restricted to direct truth-telling mechanisms.
Using
this result,
we prove that
the provision of incentives to politicians can be separated from the provision of incentives
to,
and
from redistribution, across individuals. This also enables us to develop a method of characterizing the
best sustainable
mechanism
to additional political
variables.
as
a,
solution to a sta.ndard dyna.mic
mechanism design problem subject
economy and commitment constraints formulated only
as functions of aggregate
Using this formulation, we provide conditions under which distortions created by
economy and commitment problems
are as patient as (or
more
persist or disappear in the long run.
pa.tient than) the citizens, these distortions
In particular,
if
political
politicians
disappear asymptotically, and
they remain positive otherwise. Finally, we extend our analysis to the case where the government cares
both about
results to
its
own consumption and
the future utility of the citizens. This extension generalizes our
environments where the key constraint
is
the time-inconsistency of a (partially) benevolent
government.
Keywords: dynamic
incentive problems,
mechanism
design, optimal taxation, political economy,
revelation principle.
JEL
Classification: Hll, H21, E61, P16.
'We thank
tance.
audiences at
many
.seminars for helpful
comments and Oleg Itskhoki
for excellent research assis-
Introduction
1
The major
insight of the optimal taxation hterature pioneered
by Mirrlees (1971)
is
that
the tax structure ought to provide incentives to individuals to work, exert effort and invest,
This insight
while also providing insurance.
taxation literature,
assuming that
The optimal
is
dynamic
also central to the recent optimal
This literature characterizes the structure of optimal (nonlinear) taxes
by a benevolent government with
policies are decided
full
commitment power.
tax structure typically involves a significant amount of information gathered in
the hands of the government as well as a range of transfers to and from the government using
the available
who
fiscal
instruments. In practice, however, tax structures are designed by politicians,
to future policies or to
dynamic mechanisms.
famous quote by Juvenal, which
guard the guardians?"
will
more challenging because
in their
.
much
In such environments
of the
amount
commit
This observation gives greater weight to the
at the root of
is
individual biases and cannot
of political
economy
analysis:
''who
"guarding the guardians" becomes even
of information
and enforcement power concentrated
—the government's- -hands.
In this paper,
constraints
also
own
care about reelection, self-enrichment or their
—that
we study the structure
is,
when there
is
of
dynamic nonlinear taxation
no commitment
to policies
inider political
and the government
economy
(politicians)
need to be "guarded" (controlled).^ Although implications of the self-interested behavior
of politicians for feasible optimal policies has not received
troduced by lack of commitment
Chari and Kehoe (1990),
for
in the design of
much
attention, the difficulties in-
optimal policies are generally well- recognized.
example, provide a comprehensive discussion of the implications
of incorporating time-consistency constraints.
The
challenges created by time inconsistency in
are especially severe
and were
first
example economy, where, similar
pointed out by Roberts (1984).
benchmark Mirrlees model, the economy
Under
full
Roberts considered an
to Mirrlees (1971), risk-averse individuals are subject to
unobserved shocks affecting the marginal
persistent types.
dynamic nonlinear taxation environments
is
disutility of labor supplj^.
repeated
T
But
differently
from the
times, with individuals having perfectly
commitment, a benevolent planner would choose the same
alloca-
tion at every date, which coincides with the optimal solution of the static model. However, a
benevoleiit government without
tion that
it
full
commitment cannot
refrain
from exploiting the informa-
has collected at previous dates to achieve better risk sharing ex post. This turns
the optimal taxation problem into a dynamic
game between the government and the
citizens.
'
Since in the literature following Mirrlees the optimal tax-transfer program is a solution to a mechanism design
problem, we use the terms "optimal tax-transfer program," "optimal nonlinear taxation" and "mechanism"
interchangeably.
T
Roberts showed that as discounting disappears and
of this
game
worst type at
invoh'es the highly inefficient
all
outcome
^ oo,
in
dates, supply the lowest level of labor
the unique sequential equilibrium
which
all
individuals declare to be the
and receive the lowest
level of
consump-
This example not only shows the potential inefficiencies that can arise once we depart
tion.
from the unrealistic case of
lights that the
principle,
may
main
also
full
commitment, even with benevolent governments, but
tool of analysis in
fail (in
dynamic taxation problems, the celebrated revelation
Roberts's economy there
is
no truthful reporting of types).
In light of this stark difficulty highlighted by Roberts (1984),
structing equilibrium taxation policies in the presence of political
there any hope of con-
is
economy and commitment
among
constraints that can provide incentives to and redistribution (risk sharing)
The main
in Mirrlees's baseline analysis?
contribution of the current paper
under reasonable assumptions, taxation and redistribution
from the normative Mirrleesian
also high-
analj'^sis
is
agents as
show
to
that,
policies resembling those resulting
with commitment and a benevolent planner can be
supported as equilibria. To present our main results in the clearest possible way, throughout
the paper
we
focus on the equilibrium of the
dynamic game between
and
citizens
politicians
that maximizes the ex ante utility of the citizens, and refer to this as the best sustainable mech-
anism. This terminology emphasizes that we are characterizing the best tax-transfer scheme
that
is
sustainable in the sense of being incentive compatible both for the citizens and for the
politicians entrusted with
Two
policies.
economy
ingredients are essential for our approach. First, instead of a finite-horizon
as in Roberts,
we consider an
use standard repeated
Roberts.
implementing the
game
infinite-horizon environment.
This makes
it
possible for us to
strategies to sustain better equilibria than those
emphasized
Second, we choose a particularly tractable model of political economy and com-
mitment problems.
Specifically,
we assume that
politicians have
no commitment power and
can even deviate from their within-period commitments, but they are subject to electoral
countability.
These assumptions are similar to those made
economy based on the approach
first
proposed
bj'
in the baseline
models of
is
4, for
a
modern exposition and
taxation in the presence of political economy and
political
their policy
commitment
office (see
references).
These two ingredients enable us to develop a tractable framework
is
if
not in line with the electorate's expectations, then they can be voted out of
Persson and Tabellini, 2000, Chapter
ac-
Barro (1973) and Ferejohn (1986). In the
Barro-Ferejohn model, politicians can choose any policy vector they prefer, but
choice
in
for analysis of
constraints.
Our
dynamic
first
result
that because of the potential electoral controls on politicians, a version of the revelation
principle, truthful revelation along the equilibrium path, applies in our
environment regardless
of the discount factors of various parties (that
It
this
is,
is
not a folk theorem type result).^
should be noted that this result does not generalize to other dynamic mechanism design
problems and exploits the
Our second
result
is
the presence of political
able
specific structure of
our economy.'*
key to a tractable analysis of dynamic nonlinear taxation problems in
economy and commitment
mechanism enables a separation between
constraints:
we show that the best
private and public incentives
— that
is,
sustain-
incentive
compatibility for individuals can be treated separately from ensuring that the politician in
power does not wish
to deviate
More
specifically, this
nism
in
1.
two
We
of
from the candidate
social plan (proposed tax-transfer scheme).
separation result enables us to characterize the best sustainable mecha-
steps:
first
solve the problem of providing incentives to individuals given aggregate levels
consumption and labor supply.
We
call this
a quasi- Mirrlees problem as
it is
a usual
dynamic Mirrlees problem with two additional constraints on aggregate labor and consumption.
Its solution leads to
an indirect
as a function of the aggregate levels of
2.
We
utility
functional representing expected utility
consumption and labor supply.
then provide incentives to politicians by choosing aggregate variables and the level
of rents paid to the politician.
This formulation not only provides us with a tractable strategy
for characterizing the best
sustainable mechanism, but also enables a direct comparison between the best sustainable
mechanism and the full-commitment Mirrlees mechanism
in
terms of the aggregate distortions
caused by the former relative to the full-commitment Mirrlees allocation. This result, therefore, implies that
incorporating lack of commitment and self-interest of politicians does not
necessarily invalidate the
dynamic mechanism
Our main
evolution.
methodology
design;
results focus
First,
it
dynamic taxation problems
as one of
simply adds additional constraints on aggregates.
on the characterization of these aggregate distortions and their
we show that
duce further distortions
of approaching
political
in the sustainable
economy and commitment problems always
mechanism
relative to the
intro-
full-commitment Mirrlees
^One can always construct an extended game in which there is a. fictional disinterested mechanism designer,
with the government as an additional player that has the authority to tax and regulate and the ability to observe
all the communication between the fictitious mechanism designer and individual agents.
Although a version
of the revelation principle would apply in this extended game, this docs not circumvent the substantive issues
raised here: the party entrusted with taxes
nor
and transfers has neither the same interests as those of the
citizens
much commitment power.
There has recently been important progress in the characterization of dynamic mechanisms without commitment. See, in particular, the important work by Skrcta (2000, 2007). Sloct and Ycltckin (200C) provide a proof
of the revelation principle for sufficiently high discount factors in a dynamic economy with time inconsistency,
but without
political
economy
constraints.
mechanism.
Intuitively,
if
the sustainabihty constraint of the pohtician were always slack,
then the politician would receive zero consumption and would find
expropriate some of the output.
any increase
If,
beneficial to deviate
and
on the other hand, the sustainability constraint binds, then
output must be associated with increased rents
in
it
for the politician in
power.
This in turn increases the opportunity cost of production and leads to a reduction in labor
supply and capital accumulation. Therefore, labor supply and capital are depressed in the best
mechanism
sustainable
as a
way
of relaxing the sustainability constra.mt (and thus reducing
the rents allocated to the politicians in power).
Second, we provide precise conditions under
economy and commitment
distortions disappear or persist over time. In
which these
particular,
political
we show that when
politicians are as patient as
additional distortions created by political
tion of resources converges to that of a
good spending. In
of public
(more patient than) the
economy disappear
in
citizens,
the long run and the alloca-
dynamic Mirrlees economy with an exogenous
this limiting equilibrium, there are
no additional taxes on labor
beyond those implied by the optimal Mirrleesean taxation and no aggregate taxes on
tal.^
In contrast,
when
level
capi-
politicians are (strictly) less patient than the citizens, the structure
dynamic Mirrlees economy and features additional labor
of taxes never converges to that of a
and capital taxes even asymptotically. This
last set of results is
important, since
it
provides
an exception to most existing models, which predict that long-run taxes on capital should be
equal to zero
is
(cf.
footnote 4) and might provide a possible perspective for
why
capital taxation
pervasive in practice.
The
final set of results focus
on politicians that are
main constraint on the form
the
of taxation
than the self-interested behavior of the
show
commitment
politicians.
enables us to revisit the stark negative result
We
is
in
at least partly benevolent. In this case,
This case
to the sequence of policies rather
is
particularly important since
Roberts's (1984) seminal paper discussed above.
that, in contrast to Roberts's analysis, aggregate distortions created by
problems disappear
Our paper
acterizing
is
if
the politician
related to a
is
number
of different literatures.
The
relationship to papers char-
dynamic mechanism design problems without commitment, such
the important paper by Bisin and Rampini (2005),
the presence of
ameliorating the
This result
models,
e.g.,
is
commitment
as patient as the citizens.
2007) or Sleet and Yeltekin (2006), has already been mentioned. Our paper
how
anonymous markets
who extend
as Skreta (2006,
is
also related to
Roberts's analysis and show
acts as an additional constraint on the government,
commitment problem. The
role of
anonymous markets
in Bisin
and Rampini's
therefore similar to that of zero limiting taxes on capital in the first-generation
Chamley (1986)
it
or .ludd (1985), but
is
Ramsey-type
derived here without any exogenous restriction on tax
instruments (sec Kochcrlakota, 200-5, for the zero capital tax result using the second-generation approach).
It is important to emphasize, however, that this limiting allocation can be decentralized in different ways,
and some of those may involve positive taxes on individual capital holdings.
analysis
is
related to the lack of
The most important
commitment by
the self-interested government in our model.
distinction between the two approaches
is
the infinite horizon nature of
our model, which enables us to construct sustainable mechanisms with the revelation principle
holding along the equilibrium path.
This enables us to analyze substantially more general
environments, and to characterize the limiting behavior of distortions and taxes.
In addition to these papers, our work
political economy,''
rates the general
and Tsyvinski
is
related to the burgeoning literatures on
dynamic
and on dj-namic nonlinear taxation. In particular, our framework incorpo-
model of dynamic nonlinear taxation considered
(2003), Kocherlakota (2005),
and Albanesi and
in Golosov, Kocherlakota,
Sleet (2005).
A recent
interesting
paper by Albanesi and Armenter (2007) studies general structure of intertemporal distortions
and uses a technique
in a varietj' of contexts
similar to those in this paper in separating
aggregate from idiosyncratic distortions.
The
results in this
paper are also closely related to our previous work, Acemoglu, Golosov
and Tsyvinski (2006, 2007a, b). Many of the
results here were first presented in the
paper, Acemoglu, Golosov and Tsyvinski (2006).*'
developed
in
A
working
special case of these results have
been
Acemoglu, Golosov and Tsyvinski (2007a). That paper focuses on the problem
of controlling a self-interested politician in a representative agent neoclassical growth model,
and thus does not feature any incomplete information, heterogeneity or nonlinear taxation,
which are our
pi'esent focus.
more general than those
in
Consequently, the results in the current paper are significantly
Acemoglu, Golosov and Tsyvinski (2007a) and enable us to
vestigate questions related to the structure of taxation under political
economy
in-
constraints.
In particular, the key results of the present paper related to truthful revelation, separation
of private and public incentives, the structure of nonlinear taxes and the interaction between
nonlinear taxation, political economy and time- inconsistency are not present in that paper.
The main
is
parallel
between the current paper and Acemoglu, Golosov and Tsyvinski (2007a)
the emphasis on the relative discount factors of politicians and citizens.
incentives to politicians in our
model
dynamic principal-agent analyses
1981, Ray, 2002).
Ray
is
(see,
The
provision of
also related to the structure of optimal contracts in
among
others, Harris
and Holmstrom, 1982, Lazear,
(2002) provides the most general results in this context.
Acemoglu,
^For general discussions of the implications of sclf-intercstcd behavior of governments, petitions and bureau-
among others, Buclianan and Tullock (1962). North and Tlioma.s (1973), North (1981), Olson (1982),
North and Weingast (1989), and Dixit (2004). Austcn-Smith and Banks (1999), Persson and Tabellini (2000)
and Acemoglu (2007) provide introductions to various aspects of the recent developments and the basic theory.
For dynamic analysis of political economy, focusing mostly on Markovian equilibria, see, among others, Krusell
and Rios-Rull (1999), Acemoglu and Robinson (2000), Hasslcr ct al. (2005), and Battaglini and Coate (2006).
"Results related to the comparison of market-based and government-controlled allocations in that working
paper have been extended in Acemoglu, Golosov and Tsyvinski (2007b). None of these results are present in
crats, see,
the current paper.
Golosov and Tsyvinski (2007a) extend Ray's results to the case in which discount factors are
different
The
between the principal and the agent
rest of the
paper
vironment. Section
politician).
organized as follows. Section 2 introduces the general economic en-
is
3 describes
economy environment. Section
ful revelation
and the
(or the citizens
how
the tax-transfer mechanisms are designed and the political
economy there
4 establishes the result that in our
will
be truth-
along the equilibrium path and allows us to focus on direct mechanisms (subject
to the relevant incentive compatibility constraints).
how
Section 5 shows
the provision of in-
centives to individuals can be separated from the provision of incentives to politicians, enabling
a relatively tractable analysis of sustainable tax-transfer mechanisms. Section 6 contains our
most general
from
result, characterizing the additional distortions arising
political
and commitment problems and how these disappear asymptotically when
patient as (or more patient than) the citizens.
and additional
results in
politicians are as
Section 7 provides a tighter characterization
two special cases (private
and constant types). Section 8
histories
provides a numerical example for a general best sustainable mechanism and illustrates
tortions created
by
political
have the same discount
economy can disappear quite quickly when
factor.
the government (politician)
is
economy
politicians
how
and
dis-
citizens
Section 9 generalizes our results to an environment in which
partly benevolent, deriving utility from the expected utility of
the average individual in the society. 'This model enables us to relate our findings more clearly
to Roberts (1984) as well as to the literature on the ratchet effect
and
Tirole, 1985,
(e.g.,
Freixas, Guesnerie
and Bisin and Rampini, 2005). Section 10 concludes, while the Appendices
contain additional technical materials and proofs omitted from the text.
Model
2
We
consider a general dynamic Mirrlees optimal taxation setup in an infinite horizon economy.
There
1,
by
is
The instantaneous
/.
where
a continuum of individuals and
cj
>
is
we denote the
utility function of individual
the consumption of this individual,
This formulation
individual's "type".
is
I]
set of individuals,
f
G
G /
at
[0,r|
time
is
/,
is
which has measure
given by
labor supply, and
dl is
the
general enough to nest both preference shocks and
productivity shocks.^
Let
G =
{9o,9i,
...,0j\[}
with the convention that
is
the worst type.
6i
be a
finite
ordered set of real numbers denoting potential types,
corresponds to "higher
skills"
than
0,:_i,
and
in particular, Oq
Let Q-^ be the T-fold product of Q, representing the set of sequences of
^For example, productivity shocks would correspond to the case whore u
(cj,
/J;
|
0\)
—
u {clJl/9l).
T -
length
type sequence
of individual
time
to
We
i.
6^°°
drawn from G°° according
as
from 0'^.
i
The
0'''
only knows
t.
No
All individuals have
economy
lifetime type sequence
from 0°° according to the same measure
We
6*''°°
be the draw
of this individual at
skill level
to denote the history of this individual's skills
other agent in the
assume that each individual's
type distribution.*
the
Q\, is
Let
.
up
and make the standard measurability assumption that the individual
t,
at time
6''''
think of each agent's lifetime
some measure [v^
to
6'''°°,
t-th element of
use the standard notation
and including time
We
with each element belonging to 0.
l,2,...,cx),
/i°°,
will directly
drawn
is
so that there
is
denote the distribution of the vector
same discount
factor
G
/3
and independently
identically
no aggregate uncertainty in the
0*
across agents by G*.
thus at time
(0, 1),
We
observe this history.
they maximize
i,
oo
E
/?^u(cj+„/j+J^J,
.s=0
where
E
denotes the expectations conditional on having observed the history
[-1^''']
impose the following standard assumption on the
Assumption
1
(utility
function) For
all 9
in
utility hmctioii.
G Q, u
and
ously differentiable and jointly concave in c
I.
We
0*'*.
(c,
a.nd is
I
\
0)
:
M+ x
[O,
— R
>
/]
non- decreasing in
i5
twice continu-
and non-increasing
c
I.
The production
side of the
economy
described by the aggregate production function
is
Y = F{K,L),
where
K
is
capital
>
stock, A'o
Assumption
tiable in
scale,
L G
K
and
[0,l\.
and L
=
is
labor,
and the economy
In addition,
i
2
(production structure)
and L with
satisfies
F
is strictly
full
partial derivatives denoted by
=
Win i^qFi{K,L)
oo for
K
all
Fk and
>
and
L 6
[O, r|
fined
hy
Y—F
Fi, exhibits constant returns to
0.
depreciation assumption and the assumption that labor
is
implies that there
(F,
l)
capita (and thus the
e
(0, oo),
maximum
is
a
The condition
maximum
where
that lini/v— <do
/
is
total labor supply).
the
Fk
[K, L)
maximum amount
The condition that
1
for
all
essential for produc-
steady-state level of output that
recall that
<
lini/^^oo F/^ (A', L)
F {K,0) —
tion are adopted to simplify the notation.
with
of capital
increasing and continuously differen-
Moreover, capital fully depreciates after use, and
Both the
endowment
we assume:
at
0.
starts with a positive
is
<
1
together
uniquely de-
of labor supply per
lim/^_o
Fl [K, L)
=
oo
implies that in the absence of distortions there will be positive production.
^This structure imposes no restriction on the time-series properties of individual
independent draws and arbitrary temporal dependence are allowed.
skills.
Both
identical
3
Political
Economy
Basics
3.1
The
allocation of resources in this
economy
is
who
entrusted to a politician,
is
in
charge of
operations of the government. This politician has the power to tax and redistribute resomrces
across agents, and can also allocate
We
consumption).
some
of the tax revenue to himself as rents (government
interpret the government (and thus the politician) as being necessary for
the operation of the tax-redistribution mechanisms (as well as other functions, such as imple-
mentation of law and order and provision of public goods). The key dilemma facing the society
is
how
to control the government, once the powers to tax and redistribute resources have been
vested with
We
it.
adopt the simplest and most conventional approach to
classic electoral accountability setup of
In particular, there
is
where x denotes the
neous
from that of the
distinct
Barro (1973) and Ferejohn (1986) into our environment.
a politician at time
utility of
politician's
utility function.
problem, and incorporate the
a large number of potential (and identical) politicians.
by I. The
set of politicians
this
consumption
(rents), v
:
t is
given by
R+
-^
M
is
Notice also that the politician's discount factor,
citizens,
/3.
To simplify the
analysis,
from the citizens and never engage
denote the
the politician's instanta6, is
potentially different
we assume that potential
in production,
We
politicians are
and that once they are replaced
they do not have access to capital markets.''
Assumption 3
isfies v' (:r)
>
(politician utility) v
for
all
x e 1R+ and v
(0)
is
twice continuously dijferentia.ble, concave,
=
0.
Since the politician in power both lacks
priate output for
politicians as a
1990, 1993).
its
Moreover
Our purpose throughout
is
£
the ability to expro-
the interaction between the citizens and the
literature
on sustainable plans (Chari and Kehoe,
to characterize the equilibrium of this
game between
the politicians and the citizens, corresponding to the best sustainable mechanism
the sustainable
mechanism that maximizes the ex ante
^All of the results in this paper hold
if
sat-
(0, 1).
commitment power and has
own consumption, we model
dynamic game following the
5
and
— meaning
utility of citizens.'"
a politician has access to capital markets after deviation, and only
the right hand side of the sustainability constraints need to be modified.
'"Since
selection
we
are dealing with a
among
the
many
dynamic game, our focus on the best sustainable mechanism
equilibria.
is
essentially a
Alternatively, one can think of the "social plan" as being designed by
the citizens to maximize their utility subject to the constraints placed by the self-interested behavior of the
government. In addition, throughout the paper wo focus on perfect Bayesian equilibria (sec Definition
1).
The
We
of this section formally defines the structure of the game.
rest,
by citizens and the politicians, and the timing of events.
feasible actions
first
We
describe the
then provide a
formal definition of mechanisms.
Timing and Actions
3.2
We
define a
submechanism
nism between the
mechanism
t
at time
A
i)
as a
submechanism
of the overall
specifies
t,
may
liave
mecha-
what happens
at a
with a generic element
include messages about current type of the individual,
G 6'~' (even though the individual
past types 9
subcomponent
Zt be a general message space for time
let
may
This message space
Period
and the individuals.
politician
given date. In particular,
zt-
(or
in
made some
9f
G Q, and
different reports
t
about
his or her types in the past),
and might
denote a generic element of Z' and by
z^
A
submechanism
Z
also include other messages.
the space of
consists of two mappings,
i.e.,
all
Let Z^
=
Yl Zg,
s=0
such lifetime reports.
Mt =
(q,Z(J such that
ct
Z^
:
^>-
M_|_
assigns consumption levels for each complete history of messages and public histories, and
It
:
Z^ ^>
below
[O,/]
will
However, as the timing of events
assigns corresponding labor supply levels."
make
it
clear,
we
also
assume that there
is
freedom of labor supply,
that each individual can always disobey the labor supply allocation implied by
l\
=
0.-^^
message x® such that
—
if 2'
individual in question. This
(2'"^^;
is
)
for
any
G Z*~\ then
clearly without loss of
specifies
It
any generality, since
did not exist, the individual could always disobey U and choose
l\
—
if
It
and choose
we introduce a
Instead of introducing an additional action designating this choice,
2'"^
in the sense
l\
=
for the
such a message
Q himself.
We
denote
the set of feasible submechanisms, which allow for message z® specified above and satisfy the
relevant resource constraints (specified below),
The
commit
tj'pical
to a
assumption
in
submechanism
by Mt.
models with no commitment
at a given date, but
''The mechanisms wc describe here allow
is
that the
mechanism designer can
cannot commit to what mechanisms
will
be
message spaces, but impose two restrictions. First, they
the text, hi the Appendix, we consider potentially
stochastic mechanisms to convexify the constraint set. Second, a more general mechanism would be a mapping
from the message histories of all agents, not just the individuars history. Since there is a continuum of agents
that do not share any information, this latter restriction is without loss of generality here (except that off the
equilibrium path, some submechanisms would violate the resource constraint, though this is not important for
our equilibrium analysis). Notice also that while the submechanism restricts each individual's allocations to be
a function of only his own history of reports, as it will become clear below, the government's strategies allow
are non-stochastic.
submechanisms
Finally,
This
is
for general
only to simplify notation
in
to be functions of the reports of all agents in the past.
we could
define a
submechanism as a mapping Mi [/<;,] conditional on the capital stock of the economy
what can be achieved will be a function of the capital stock. Wc suppress this
at that date to emphasize that
dependence to simplify notation.
In Acemoglu, Golosov and Tsyvinski (2006), wc derive the "freedom of labor supply" endogenously using
an environment in which individuals could be disabled and unable lo supply any labor. Directly introducing
the freedom of labor supply is without loss of generality for our main focus and simplifies the analysis.
'
A
offered in the future.
an additional type of deviation
for the politician in
extract resources from the society even within the
We
time
t,
next summarize the
the
economy
game between
A'/.
At the beginning
of period
t,
2.
Individuals send a message
z]
i.
£
At
is
period.
power and the
in
power and a stock
in
the politician offers a
e Zt.
The message
g ^t-i determine labor supplies
indexes individuals and
[0, 1]
same
I
G
that there
citizens.
At each
of capital inherited
Then:
1.
i
i.i
is
power whereby she can use her power to
the politician
starts with a politician
from the previous period.
^i,t-i
economy context
natural assumption in the poHtical
this point, individual
i
2'^*
//,
z,'
submechanism Mi G Mttogether with the history of messages
according to the submechanism Mj, where
(z'"')
e Z^ denotes the history of reports by individual
can also choose to supply zero labor and receive zero
consumption.
3.
4.
Production takes place according to the labor supplies of the individuals, with Yt
—
F{Kt,Li), where Kt
—
The
is
politician decides
{0,1}.
=
If ^f
0,
the capital stock inherited from the previous period, and L(
whether to deviate from the submechanism Mt, denoted by
production
submechanism Mi € Aii, the
stock
is
determined as
chooses x[
< F
capital stock
5.
is
A'f'_,_i
among
politician chooses it
F {Ki, Li)
A'/+i --
{Kt.Lt), and a
is:
distributed
-
it,
—
new consumption
= F {K'l, Lt) —
i[
-
j^^^
^j
G
agents according to the pre-specified
< F
{h'l, Lf),
J,^^j Ct
(;'') di.
function
c'l
:
and next period's capital
If ^,
=
1,
the politician
Z' -^ IR+, and next period's
c[ (z'-') dz.^''
Elections are held and citizens jointly decide whether to keep the politician or replace
him with a new
one, denoted by
by V} G
the set of
{0, 1}'
the set of
all
all
p,,
G
{0, 1},
where
p^
=
I
denotes replacement. Denote
possible histories of electoral decisions at time
possible electoral decisions. Replacement of politicians
is
t
and by
TZ
without any costs.
Note the difference between the standard models with no commitment and our setup,
where, in stage
and
4,
the politician can decide to expropriate the output produced in the economy,
citizens can replace the politician at the last stage.
decisions have already been
made according
Notice that at stage 4 labor supply
to the pre-specified
submechanism Mt- However,
''More generally, we can allow the govcinniont to capture a fraction r; < 1 of the total output of the
^ = 1, whore the level of r; could be related to the institutional controls on government
or politician behavior.
In this case, the constra.int on the government following a deviation would be
= i]F {Ki, Li) - x[ — Jii^jc't (.5'') di, with the remaining 1 - 7; fraction of the output getting de/("('.(-i (/i')
stroyed, This generalization has no effect on our results, and we set r/ = 1 to simplify notation.
economy when
10
—
consumption allocations cannot be made according to Mt, since the
some
consumption allocation function,
The important
This
jointly.
is
cj
make
— R+
Z'
:
feature of stage 5
cisions independently, they
ment
we
of the output for herself. Consequently,
>
at this point.
that even though individuals
is
is
no
expropriating
is
power choose a new
also let the politician in
their political decisions
natural since there
politician
make
their
—elections to replace the politician
conflict of interest
decision. Joint political decisions can be achieved
among
the citizens over the replace-
by a variety of procedures, including
various voting schemes (see, for example, Persson and Tabellini, 2000, Chapter
simplify the discussion by assuming that the decision
citizen.
is
taken by a randomly chosen
and Reporting Strategies
M — {Mt}'^(^ with Mt G M.t
Let
{0, 1}
Here we
4).
^^
Histories
3.3
G
p,
economic de-
be a mechanism
above), with the set of mechanisms denoted by
politician's
consumption
We
levels (rents).
(i.e.,
M.
a sec^uence of submechanisms defined
—
Let x
be the sequence of
{.T/}J^g
define a social plan as (A/, x), which
an implicitly-
is
agreed sequence of submechanisms and consumption levels for the politician.
We
represent the action of the politician at time
elem.ent of vt
second
is
the
submechanism that the
herself
citizeiis,
if ^^
—
0.
Since
fourth element, x[,
is
Finally, the fifth element
is
original
specifies
(politician)
the function
Cj
third element of vt
is
time
c\
A','_|_j,
is
consumption
t,
what the
first
and the
politician
both total production and total consumption by
is
determined as a residual from
politicians.-'^
when
^^
= L
that the politician chooses after deviating from the
denoting the set of
but to simplify notation we write
all
such functions. Once again the capital
determined as a residual from the resource constraint.
levels
Xf, x[
must
satisfy: xi
< F
6 K+. Let Tt be the
the history of D/'s up to and including time
=
at stage 1 of
the consumption level for the politician in power
stock for the following period,
Let h^
off'ers
The
{Mf,^i,Xt,x'i,c'A.
not specified as part of the action profile of the
is
submechanism, with
Government
Mt
The
=
Vj
given xt the capital stock for next period, A'(+i,
the resource constraint and
The
politician
the politician's expropriation decision.
consumes
the
is
by
/,
t,
and assume that
(Kt, L))
and
set of vt's
and
this
is
x[
f*
< F
{Kt, Lt),
£ T* denote
publicly observable.-'^
{Ko,io,V(),fjQ,Ki, ...Kt,,ii,vt, Pi,Ki^-i) denote the public history of the
game up
Exactly the same equilibrium is obtained if tlierc arc majoritarian elections over the replacement decision
and each individual votes sincerely or uses strategics that are not weakly dominated in the election.
''Since we are characterizing a (sustainable) mechanism, there is no need to specify the ownership of the
capital stock Kt+i. Instead, this is simply the amount of resources used in production in the folio-wing period,
and the government (the mechanism) decides how this production will be distributed,
In fact, v'' includes the action .xj and the function cj, which are not observed when E,, = 0. Thus, more
appropriately, only a subset of v' should be observed publicly. This slight abuse of notation is without any
^''
consequence
for tlie analysis.
11
to date
and
t,
be the
Fl'
The
set of all such histories.
electoral decision at time
t,
p^, is
defined
as
://'-^ X(^, -.{0,1},
p,,
given the public history at time
whether to replace a
when
her type history
history
up
to time
t
—
h''^ The action
are
1
8*.
a
/-I
^t
I
h'-^
zt [9']
i''-'"^
I
:
Z'-^ X
if it
t,
the society chooses
)
as the reporting action of
far
aj specifies a
jy-'-i
X G* ->
an individual
=
=
[0'"'']).
zt [9']
for all 9*
means that the
message
£
^(
Z(, so:
Zf
q-(
for
an agent of type
G Q'\
z'-^ e
individual
notation,
Z*-\ and
H'-\
h'"^ G
where
recall that
sending a message that fully
is
we represent the
truth-telling strategy
Let
We
2j
E Zt he
In addition,
let
by
,
we have
us define the null strategy
;® for all 0'
S S'
,
z'^'
G Z'-\ h'-' £ H'-\
stands for the message corresponding to zero labor suppl)'.
~^'~',
\
Si
h'~^
)
=
a® to denote that the individual
profile of reports at time t}'^
denote the reporting strategy
sponding to the
re-
I
,~®
the notation a\ {9%
(2)
Notice that this strategy only imposes truth-telling following
Q:*,
z'-\h'-') =
,
at time
and the publicly observed
is z''^'^
truthful reports in the past (because instead of an arbitrary history of messages z^~^
conditioned on z'"^
i
satisfies
To economize on
veals her true type.
z^~^ [^'^^]
<-i
,
,
where the notation
a] i9t
at time
denote the message resulting from strategy
truth telling
is
j
9\ her history of messages so
is
(at (6'*)) to
z''
and actions of politicians
^'"^ /V^'
\
al
We write
A strategy
1
politician.
For the citizens, define a] (9^
t
-
/.
set of all
As
usual,
we
is
We
will use
playing the null strategy.
define
profile of all the individuals in society
such reporting strategy profiles.
We
by a^^ with
,
A
corre-
denote the strategy profile of
all
the individuals in society by {a, p}.
''More formally,
i
6
i
G
[0, 1]
"More
[0, 1]
s,
assigns a report to each individucil, ihus
denotes individual
/,
and 2/
is
the set of
all
formally, a, assigns a report to each individual, thus
denotes
individuii.l
it is
a
function of tho form z
[0, 1]
:
—
Zt,
where
such functions,
i.
V2
it is
a function of the form
a
:
[0, 1]
-^ Z, where
Definition of Equilibrium
3.4
The
strategy of the poHtician in power at time
Ft
tliat is, it
determines
Mt e Mt,
pubhc
as a function of the
e
?<
liistory
t
is
therefore
H*"^ X Z'-i -^ T,
:
G
{0, 1}, xj
[0,
F [Kt, Lt)\,
and the entire history
strategy profile of the politician's by
and the
1"
1
A (Perfect Bayesian) equilibrium
the citizens
is
given by strategy profiles
p}
sequentially rational,
Q'^e
F and
{«, p}-
i.e.,
We
by
F (A',, L,))
citizens.
is
We
cj
G
C*
denote the
by Q.
game between
the politicians
and
F and
a belief system B, such that
best responses to each other in all
B
and
information
sets,
derived from Bayesian updating given
write the requirem.ent that these strategy profiles are best
responses to each other as F t:{Q.p} ^
p'
in the
F and {a.p} and
given B, and whenever possible, the belief system.
the strategy profiles
of reports
[0,
set of these strategies
Definition
{q.:
e
S:[
.for
a.ll
F ^ Q and {a.p} bf
{»': p'} for all
a's
A
and
£71.
In what follows, there will be no need to explicitly characterize or condition on the belief sys-
tem B (though
this
is
always in the background). Let us define
as the action profile of
Definition 2
{^> p]
M
is
tlie
politician induced
a sustainable
for the citizens and a strategy
equilibrium and induce an action
Mt — Mt
profiles
,
^t
Fm,x
~
0,
o.nd
ci.nd
Xi
=
x-t
profi.le
profi.le
for
by strategy F given a
mechanism
all
h''
<
if
=
rjv/,,T
there exists x
\
Mt^
^j, xj, x^, cj
social plan
=
>
(M, x).
{xjIJ^q, a strategy profile
Fm,x G G for the government, which constitute an
i\//,^j,.x't,Xj, cj
G H^. In
for the politicians such that
\
this case,
{a,p} support the sustainable mechanism
we say
that equilibrium strategy
M.
In essence, this implies that the politician in power does not wish to deviate from the social
plan (M, x) given the strategy
profile,
{a.p}, of the citizens.
The notation F
t{Q,p}
F makes
this explicit, stating that given the strategy profile, {a, p}, of the citizens, the politician
prefers this strategy profile to any other strategy profile based on the
4
The
implicit agreement.
Truthful Revelation Along the Equilibrium Path
is
a powerful tool for the analysis of mechanism design and implemen-
e.g.,
MasCollel, Winston and Green, 1995). Since, in our environment,
revelation principle
tation problems (see,
the politician in power,
ests
same
weakly
than those of
tlie
who
operates the mechanism, cainiot commit and has different inter-
agents, the simplest version of the revelation principle
13
may
not hold; as
Roberts's (1984) paper discussed in the Introduction demonstrates,-' ° there
in
may
exist situations
which no equihbrium would involve individuals reporting their true type.""
The key
be that along the equilibrium path, a version of the
result of this section will
revelation principle will hold (without introducing a fictional
positive discount factors).
The main
difference between our approach
dynamic mechanism design without commitment
sibility that the
mechanism designer and
(e.g.,
with the ratchet
and the
effect)
is
for all
literature
on
that the pos-
agents can punish the deviating politician (mechanism designer) by replacing
him. Such punishments are natural in the context of political economy models, though they
mechanism design problems without commitment. Another
are typically not present in other
important difference
is
that, as
posed on deviating politicians
will
it
will
become
clear below, the
punishments that can be im-
be independent of the history of mechanisms to date. These
differences are responsible for truthful revelation along the equilibrium path in our model.
4.1
We
Truthful Revelation
focus on (Perfect Bayesian) equilibria that maximize utility of the citizens, which
As we
to as the best sustainable mechanism.
mechanisms
(i.e.,
the constraint set, (4)-(6))
will see
is
we
refer
below, as long as the set of sustainable
noriempt}', this
is
eciuivalent to choosing the
best sustainable mechanism, given by the following program:
MAXq;
_
ma.x
E J2p*u(c,{z'[a,{9')])J,{z''[a,{9')])\9l
^^
{d,{-).lii).xi,f<i.-,i}Zo
subject to an
initial capital
(3)
L(=o
stock A'o, the resource constraint,
= F[Kujk{z'[a,{9')])dG'{e*)^
Kt+i
(4)
- jh{z*[ai\9')])dG'[9')-iu
a set of incentive compatibility constraints and electoral decisions for individuals,
{a,/?}
and the sustainabihty constTaint
Yl5'v{xt+s)
Ls=o
is
a best response to Fm.xi
(5)
of the politician in power;
>
max
r'
K'
IE
[v{x[)+5vl[k[^,A\M')]],
(6)
('
''See also Bestcr and Strausz (2001) and Skrcta (200(i, 2007)
lur
the
more general use
of the revelation
principle.
^"As noted in the Introduction, this statement refers to the case in which messages are sent to the politician in
charge of the mechanism. It is possible to construct alternative enviromnents with fictional mechanism designers
with full commitment power, so that the revelation principle holds.
14
for all
>
t
The
t:
0.
last constraint, (6),
the left-hand side
encompasses
what the
is
all
the possible deviations by the politician at date
from date
politician will receive
The right-hand
the implicitly-agreed consumption schedule for herself.
she can receive by deviating.
subgame
of the
at time
for individuals,
submechanism
^(
=
1,
at time
=
^,
i
+
1
utility,
=
and a choice
I,
maximum
the
is
new consumption schedule
together with a
from
of xt different
Xt;
(encapsulated into the continuation value
the politician chooses x\, K[_^y and
by current
side
potential deviations include a deviation at the last stage
to expropriation, ^^
t
or
c[\
The
onwards by sticking with
t
In the case where
u^).
maximize her deviation value, which
to
cj
v{xt), and continuation value, written as v^
(
new
or the offer of a
A'j'^_j,cJ
M*),
|
is
given
emphasize
to
that this continuation value depends on the entire history of submechanisms (thus on the
information about individual types that has been revealed so
capital stock from then on, A7+ii
satisfied, it is either
'^^
well as potentially on
because the politician prefers
=
^^
far)
up
to time
If this
Cj.
i, A'/',
constraint, (6), were not
and some sequence
of
submechanisms
or consumption levels different from (A/, x), or because the politician prefers
former case, we can always change {M,x) to ensure that
(6) is satisfied.
The
(6) is
subject to (4)-(6).
(3)
trivial allocation
Lemma
a along
C(,
and
Proof. Let
{MstXs}
p^{h')
=
1,
citizens, since
it
[a
\
constraint set
is
indeed nonempty, since the
for all parties
is
in the set.
denote a strategy profile where
a') to
(4)-
We
all indi-
then have:
u
(f
(a'c
(/V-i)
,
L,
(/V)))
for all
t
(7)
,
J
in the best sustainable
mechanism
involves no re-
initial politician along the equilibrium path, is identical to the solution of the
maximization problem in
if
q=
The allocation of resources
placement of the
G
—
i.e., £,^
1 Suppose Assumptions 1-3 hold. In any sustainable mechanism.,
necessary.
c'f
In the
1.
thus a solution to the program of
the equilibrium path and a' off the equilibrium path.
_.s=0
and
is
with zero production and zero consumption
E Y.5^v{xt+s) >
is
and
satisfy (6)
Finally', this
Let us also introduce the notation
viduals play
=
Consequently, as long as the constraint set given by
nonempty, the best allocation must
maximizing
E,t
latter,
cannot be part of the best equilibrium allocation from the viewpoint of the
involves government expropriation.
and on the
(MAXq)
with if il<[^^,c',
the susta.inabilit,y constraint (6)
{A'l,
xtj^Q be a solution
= {Ms.Xs] for all s <
if h' — h' and p^{h.') =
t.
to
is
M') =
\
for
all
A/' G A1^, A'^'+i
&
^-\-
equivalent to (7).
(MAXq).
Introduce the following notation:
/i'
—
h}
Consider the strategy profile p^ for the citizens such that
1 if /i*
^
h^.
15
That
is,
citizens replace the politician unless
the politician has always chosen a strategy inducing the allocation {M/,,,zv}^o i"
periods. It
a best response for the politician to choose {Mt,Xt}j^Q after history h^ only
is
E J2S\>ix<+,]
>
(c[,
If (7) is
=
(^i
/i*,
is
1
F{Kt,
This deviation payoff
Lt).
(7).
h\ the best deviation
history
is
greater than
its
for the pohtician
is
equilibrium payoff following
This contradicts sustainability and establishes that
(7)
necessary in any sustainable mechanism.
see that (7)
is
sufficient for the best sustainable
mechanism, note that reducing v^ (/\(+i,
equivalent to relaxing the constraint on problem (3), so
sumption
is
x'f
r'
some public
violated following
—
K'
the politician's continuation value following a deviation to a feasible
is
given by the left-hand side of
To
is
and
K^^i)
if
E {v{x[)+6v'f[R",^,j{\M')]
max
f'
s=0
where vf
previous
^^^
3, v^
>
achievable for
(i.e.,
M*
all
x
>
and v
€ M'\
F'
G
(0)
=
0),
is
always preferred. Since from As-
we only need
to
show that
K+ and
£
Cj.
G, A','+i £
rium strategy p^, which involves replacing the
value of the politician
establishes
tire
is
clearly
v"^
—
Cj
c^
politician
v^
(Kf^i,c[
\
MM
Under the candidate
when she
=
equilib-
deviates, the continuation
regardless of the history of play
up
to this date. This
sufficiency of (7).
Next suppose {A/(,,zv}^q that
is
(MAXq)
a solution to
Bayesian eciuilibrium with replacement of the
initial politician.
allocation {M/,X(}J^q such that the initial politician
is
Ijc
supported as a perfect
Now
consider an alternative
can
kept in power along the equilibrium
path and receives exactly the same consumption sequence as the new politicians would have
Since {Mt,Xt}'^Q satisfies (7) for the
received after replacement.
{Mj,X(}^o
satisfies (7) for all
involve at least
t
t
=
=
utility to
some
positive
t
for the initial politician.
consumption
for the
new
new
politicians at all
t,
Moreover, since {7lf(,it}^Q must
politicians, {il//,
Xj}^o
yields a higher
the initial politician. Thus, xq can be reduced and consumption of agents at
can be increased without violating
This proves that there
To complete the
replacement strategy
is
no replacement of the
proof,
p'^)
is
we only need
p'^
show that
to
if
^
h'
cannot be a solution to
initial politician
h'
(MAXq).
along the equilibrium path.
citizens' strategy (in particular, the
This follows by considering the following
sequentially rational.
continuation strategy for each politician;
This ensures that
(7), so {.Afj,.-rJ}^Q
,
and a—a^ are a best response
then Xg
= F {Ks, Ls)
and
E,s
=
1,
Vs >
t.
for the citizens.
This lemma uses the fact that regardless of the history of submechauisrns and the amount of
capital stock left for futin-e production, there
is
an equilibrium continuation play that replacing
a politician gives the dcviatior zero utility from
results in repeated
tliat
point onwards (which
is
analogous to the
games where the most severe punishments against deviations
16
are optimal,
|
M*
e.g.,
Abreu, 1988). This continuation play
from the imphcitly-agreed
output, Xf
— F{Kt,Lt).
politician to (7),
up
The
social plan.
mechanism, the best deviation
used as the threat against pohtician's deviation
is
implication
that, along the best sustainable
is
for the politician involves ^^ == 1
and expropriating the whole
This enables us to simplify the sustainability constraints of the
which also has the virtue of not depending on the history of submechanisms
to that point. '^^ Moreover, the
lemma
also
shows that
any sustainable mechanism
in
(7) is
necessarj'.
Next, we define a direct (suh) mechanism, as
mechanisms involve a
We
current type.
restricted
M*
:
—
message space, Zi
—
G'
>
[O,
/T]
x M+. In other words, direct
9(, where individuals only report their
denote a strategy profile by the politician's inducing direct submechanisms
along the equilibrium path by F*.
Definition 3 A strategy
we have
that a] (9*
\
profile for the citizens, a*, is
=
0'~\/i'~M
a*
.
We
write
truthful
a*—
[a*
|
if,
along the equilibrium path,
a') to denote a truthful strategy
profile.
The notation a*—
(a*
|
a')
rium path, but may play some
emphasizes that individuals play truth- telling along the equilibdifferent strategy profile, a', off the equilibrium path. Clearly,
a truthful strategy against a direct mechanism simply amounts
Let us next define c[r,Q;], /[r,a] and x[r,a]
of the agent.
consumption and labor
their reports),
and the sequence of government consumption
citizens,
information available up to time
for allocations of
1
as, respectively,
the equilibrium
supplj' distributions across individuals (as a function of the history of
egy profiles of the politicians and
Theorem
to reporting the true type
t
such that
levels resulting
from the
strat-
of these functions only condition on
all
time
t.
(Truthful Revelation Along the Equilibrium Path) Suppose Assumptions
1-3 hold and that T and [a, p] form a comhination of strategy profiles and electoral decisions
that support a sustainable mechanism.
profiles r*
and a*
—
(a*
|
a') for
Then, there exists another pair of equilibrium strategy
some
a' such that F* iiid.uces direct
induces truth telling along the equilibrium path, and moreover c\r c^
,
submechanisms and g*
=
c[r*,a*], i[r,a]
—
i[T\Q*], andx[T,o^^x[T\q*-].
Proof. Take equilibrium strategy
Then by
definition ^^
=
for all
t,
profiles
and from
F and a that support a sustainable mechanism.
Lemma
^'This statement refers to the sustainability constraint,
cations depend on the history of individual messages.
17
(7).
1,
(7) is satisfied.
Let the best response
The optimal mechanism
will clearly
make
allo-
.
of type 9* at time
4-'
,/7,'--^)
according to a be to announce Zfj
t
Denote the expected utihty of
By
.
d',r,h'-'
Now
4
and public history /i'~^ Let
under
ht-i
9 ,lt
U-1
)i
given a history of reports
(^9'-\h'~'A
i^z'f'
,
(e\h'-AY
ztx
mechanism given history
this
5'
definition of zU
> U
=
[0\h'-'j
this individual
;-,'-!
I
O'Ji
h'-"^
be
being a best response, we have
9\r,h t-i
for all
z*r (^9'\
h''^^ e Z* and
ft*"!
consider the alternative strategy profile for the pohtician P*, which induces the action
profile
such that
Mt,£,t,xt,x
<
submechanism) and
c [T*
,
a" ,h]
=
=
^(
c [r, a,h]
,
for all
l_\r*
,
a* Ji\
=
Mt = M^
i,
l_
a,h]
[F,
,
(wlrere M^*
and x
=
a,h]
[T,
a direct
is
x |r* a* ,h]
,
Therefore, by construction,
,li'-'
I
9\r,h t-i
9\
>
for all 9
Q* and
<E
h'"'^
all
e //'""^
response along the equilibrium path
strategy profile T*
a*
=
.
(8)
It-l
9\ h
9\r,h'^'
Equation
(8)
9 .h
f^*
is
the
when
as
a against
tliey play
e\T\ht-i
{a*
when
a') is a best
\
M*
mechanism
for the agents against the
same
t-i
=
implies that a*
^foreover, by construction, the resulting allocation
a') against
(a*
t
u
h1-1
and
politician
individuals play
Therefore, by the
F.
I
definition of
identical to
F being
a
sustainable,
we
liave
thus establishing that (F*,q*)
The most important
ciple
idea of the proof
ruler to
is
Two
^^1
is
as follows.
On
Now
T' ^ 0I"'
choose a' to be
for all F'
G G or that
is
that for the rest of the analysis,
we can
side of the agents.
the equilibrium path, there
is
effective
commitment
obey the mechanism. Then, one can use the usual proof of the revelation prin-
and construct
game
fo^'
mechanisms on the
truthful] reports tliat give the
path. Off the equilibrium path, one can use the
the
f^'
an equilibrium.
implication of this theorem
restrict attention to truth-telling (direct)
by the
^{a.p}
off-the-equilibrium path, which implies that F* h{a',p}
(7) is satisfied,
The
F
same
utility to the ruler
same punishment
on the equilibrium
strategies as those used in
witliout direct revelation. Therefore, the sustainability constraint
observations are worth making at this point.
is
satisfied.
truthful revelation along the
First,
equilibrium path and the politician's decision to pursue the implicitly-agreed social plan are
important to distinguish from truth
telling
and commitment
mechanism design problems. In these problems, there
telling along all paths
In contrast,
in
and there
is
unconditional
our environment, there
politician can exploit the information
is
it
tliat are
exists
present in
tlie
standard
mechanisms that induce truth
commitment
(i.e.,
again along
all
paths).
no commitment off the equilibrium path, where the
has gathered or expropriate part of the output. Re-
latedly, off the equilibrium path, non-truthful reporting
18
by
tire
individuals
is
both present and
e
H'-\
also important to ensure sustainability.
However, along the equilibrium path induced by a
sustainable mechanism, the politician prefers not to deviate from the implicitly-agreed social
plan,
and given
this equilibrium behavior, individuals
can report their types without the fear
that this information or their labor supply will be misused. Second, the results in
are not related to "folk theorem" type results. In particular,
and applies
Theorem
1 is
mechanism designer
not a limiting result
despite the lack of
commitment and
the self-interested preferences of the
(politician), a revelation principle type result holds
twofold. First, the
is
setup where the politician has a deviation within the same period ensures that
ation paj'off after deviation
If
ful reporting
make
is
tiie
would become more
is
its
continu-
independent of the information revealed along the equilibrium
the continuation value of
production,
1
for all discount factors.
The reason wh}^
path.
Theorem
politician
depended on such information, ensuring truth-
Note that deviation within the period, following
difficult.
a very natural assumption in our setup, since the politician has the power to
transfers after production
is
realized,
and thus there
is
no reason
him not
for
and take a greater
fraction of these resources for himself than specified in the
such a deviation
beneficial. Second, in our
is
involving replacing the politician.
mechanism, making
it
agreed social plan (Af,
to deviate
mechanism
model individuals can use punishment
The punishments
if
strategies
strategies of citizens support a sustainable
the best response for the politician in power to pursue the implicitly.x).
Given
this sustainability, there
is
effective
commitment on the
side
of the politician along the equilibrium path.
Finally,
Theorem
1, like
the rest of our analysis, focuses on Perfect Bayesian equilibria.
could also impose additional refinements, such as renegotiation-proofness.
the main results in this paper (except those in Section
9)
It
can be shown that
hold without any modification
Theorem
focus on renegotiation-proof equilibria. In other words, truthful revelation in
the best sustainable mechanisms characterized in Theorems
renegotiation-proof equilibria.
Golosov and Tsyvinski (2007a)
idea
is
The argument
for
and
5 can
if
1
we
and
be supported as
very similar to that developed in Acemoglu,
an economy without incomplete information.
that citizens can punish a politician
him and
is
2, 3,
One
who
The main
deviates from the social plan by replacing
following an equilibrium along the relevant Pareto frontier thereafter (thus there
no reason
for
punishments that are harmful
for the citizens themselves).
We
is
do not provide
the details here to economize on space.
4.2
The Best Sustainable Mechanism
Theorem
uals.
1
enables us to focus on direct mechanisms and truth-tehing strategy a* by
all
individ-
This implies that the best sustainable mechanism can be achieved by individuals simply
19
reporting their types. Recall that at every date, there
by G{9). This
iniplies that 9'
of G{9),
G'- [9)
often). ^^
Given
and
(since there
is
=
as Gf
that any sustainable mechanism
side of the agents,
an invariant distribution of 6 denoted
has an invariant distribut-ion, which
simply the
is
a continuum of individuals, each history
this construction,
consumption
total
is
we can write
Jq,
is
ci
{9')
total labor supply as Lt
equivalent to a direct
we obtain the main
—
occurs infinitely
Jq, k (9^)
Moreover, since Theorem
{6')"'^
dG'
9'
mechanism with
result of this section,
which
i-fold version
1
dG*
(d''),
establishes
trutli-telling
on the
be used throughout the
will
rest of the paper:
Proposition
Assumptions 1-3
1 Suppose
Then, the best sustainable mechanism
hold.
is
a
solution to the following maximization program:
MAXi
:
max
U'^^'^=
{c,(-),/,().A-,,+i.x,}^o
some
subject to
initial condition
Kq >
Kt+i
0, the
E X: /3'u(c,
((?''
),/,(0^.')
I
(9)
0i)
t=o
resource constraint
= F{Ki,L,)-Ct-xt,
(10)
a set of incentive com.paMbility constraints for -individuals,
J2 ^'''^
Q-l^t
+
f^
),k,.si9-*-'')\9U,)
i^'i+'
VJ-
(11)
oo
>
QJ,(
E j:iruU,Jt'-''),i,,J9
91t + s
,s=0
for
all
t,
all
G 0' and
<?''*
all
possible sequences of
>
<
9f_f_^
I
J
'
,
and the
sustaina.btlity constraint
s=0
of the politician
E
>v{F{Kt,Lt))
V [X t+s)
(12)
.s=0
for
all t.
The proof
Proof.
follows from
Ijemma
equilibrium (a*',r**), that nraximizes
not feature
{.,
Mt
1
\
it
—
I
for
any
t.
(9).
1
and Theorem
By
tlie
Suppose there
1.
argument
in the text,
exists
(a**,r**) will
Therefore, (a**,r**) features a sequence of submechanisms
oo
,
consumption
levels for the politician, {£f},^o ^^'^ it
'^More formally, given the
an
coiitiiiiiuni of ageuls,
wo can apply a law
=
^ for
of laigc
^1'
^-
Setting [M,x)
=
numbers type argument, and
have positive nioa-siirc. Sec, for example, Uhlig (1996).
^^From now on, we suppress the "'s to simplify notation and simply use cj, /) and xt. Note also that Jq, here
denotes Lcbesgue integrals, and in what follows, we will suppress the range of integration, G'.
each history
9' will
20
N Mi
rem
1
j-
,
{xt}.^o j implies that (a**, F**) support a sustainable mechanism. Then, use Theo-
to find (q*, r*) corresponding to a sustainable direct mechanism. This direct
mechanism
has to satisfy the resource constraint, (10), the incentive compatibility constraints of individu-
which instead of
als at all dates,
Finally,
Lemma
from
],
(5)
can be written as
(11) since F* induces direct
the constraint (12) ensures that F*
mechanisms.
a best response to citizens'
is
strategies, {a*, p}.
The
Theorem
role of
for the best sustainable
1
formulation
in this
mechanism
as a direct
is
that
enables us to write the program
it
mechanism with
truth-telling reports along
the equilibrium path, thus reducing the larger set of incentive compatibility constraints of
individuals to
Theorem
Also as noted after
(11)."'^
supported as a renegotiation-proof equilibrium
(see
1,
the result in Proposition
1
can be
Acemoglu, Golosov and Tsyvinski, 2007a).
Separation of Private and Public Incentives
5
We
next show the second main result of the paper that our analysis of the dynamic Mirrlees
economy with
self-interested politicians
is
by separating the provision
simplified
of incentives
to individuals from the provision of incentives to politicians.
Let us
first
define the
dynamic Mirrlees program (with full-commitment, benevolent gov-
ernment, and exogenous government expenditures). Imagine the economy needs to finance an
exogenous government expenditure Xt >
maximizing the time
t
=
at time
Then
t.
the d5mamic Mirrlees program of
{ex ante) utility of a representative agent, can be written as
(e.g.,
Golosov, Kocherlakota and Tsyvinski, 2003, Kocherlakota, 2005):
oo
E
max
(13)
.t=0
subject to the incentive compatibility constraints, (11), and Ct
+
Xt
-\-
Kt+\ <
F {Kt,Lt)-
Next, we add the feasibility constraint that {A'/}^q should be such that
{C,,L,}-oeA-,
where
A°°
=
{{Ct, L,,}Zo such that 3 {c, (0')
In other words, {Ct,Lt}(^o ^
{cj (^')
,lt
(^')}(^Q-
This set
A^
is
,
/,
(^')
satisfymg (11)}.
}i"o
(14)
implies that there exist incentive compatible and feasible
important to define, since, given certain government expen-
diture sequences, {XtJ^p's, the constraint set of this Mirrlees maximization problem can be
^^The equations in (11) focus on Ihc incentive compatibility constraints that apply along the equilibrium
path (expectations on both sides of the constraints are taken conditional on 0'''). This is without any loss of
generality, since (11) needs to hold for
time
t
=
is
covered by
any sequence of reports
this set of constraints,
21
{Ot^,,
\
,
thus any potential deviation from
empty
be
(e.g., if Ct.
—
and L/ >
the incentive compatibility constraints of individuals cannot
0,
satisfied).
For a sequence {C/, Lt}'j^Q e A°°, we can define the quasi- Mirriees program as
E
max
U{{Ci,Li]?Lr,)=
^/3'u(c,(0'''),/,
{cU')A()}^0
(0'-'
(15)
Lt=o
subject to the incentive compatibility constraints, (11), and two additional constraints
j
Ct {e')
dG'
{e')
< Cu
(16)
{6')
dG'
[6*)
>
(17)
and
/
It
Lt.
This program takes the sequence {C(, L/}go e A°° as given and maximizes ex ante
agent subject to incentive constraints and two additional constraints.
the
sum
of
consumption
levels across agents for all report histories to
number
Ct, wliile the second, (17), reciuires the
amount
Lt-
The
functional
U
sum
({C/, Lt}fZo) defines the
'^^^^ '^^^^
function of the individuals (from the viewpoint of time
that the functional L/({C/, L/}^o)
concave and
Returning
to the
{Xt}^Q,
this
ZY ({Cj,
we
L^J^q)
initial level of capital
Ct
+ Xt +
ex ante
{t
=
less
than some
0) utility of
an
^e interpreted as the indirect
=
0).
Appendix A, we show
In
to characterize the best sustainable
for a given
we
will
make
mechanism.
sequence of government expendi-
as;
max
subject to an
be no
allow for randomizations). In the text,
dynamic Mirriees program,
can be written
t
requires
well-defined, nondecreasing in Ct, nonincrecising in Lt,
i^
difi:erentiable (as long as
use of these properties of
tures
maximum
first, (16),
an
be no greater than some
of labor supplies to
agent in this economy for a given sequence {C/, LtY^o
utility
The
utility of
stock A'q
Kt+, <
U{{Ct.,Lt}Zo)
>
F [Kt, Lt)
(18)
and to
,
and {Ct, Lt}^o e A°°.
(19)
This derivation implies that we can represent the standard dynamic Mirriees program as
a solution to a two-step maximization problem, in whicl9 the
first
step
formulation, yielding the functional U{{Ct, Lt}^^)), and the second step
U{{Ct, Lt]fZo)
°'^''^^'
is
is
(he quasi-Mirrlees
the maximization of
sequences {C/, Lt, K't+i} subject to a resource constraint and to
This representation
is
particularly useful because
it
feasibility.
highlights the parallel between the
d3'namic Mirriees program and the best sustainable mechanism. In particular, the maximization
problem characterizing the best sustainable mechanism,
22
(9),
can be written as one of
maximizing the indicted
utility function subject to constraints:
mCuL,}r=,)
,r ."'^^';=o
(20)
subject to
Ci
+
xt
+
Kt^i <
The only
F {Ku U)
and
also subject to (12).
(19)
and the best sustainable mechanism
constraint for
tlie
difference
and {Q,
LJ^q
(21)
between the dynamic Mirrlees program
in (20)-(21)-(12) is
politician in power, (12),
G A°°,
which also makes
in (18)-
the presence of the sustainability
{xt}'f%Q
an endogenously chosen
sequence instead of the exogenously given {.Y;}^q. This formulation establishes the following
theorem.
Theorem
hold.
2
(Separation of Private and Public Incentives) Suppose Assum.ptions 1-3
Then, the
best sustainable
mechanism
solves a quasi- Mirrlees
Proof. This follows immediately from rewriting
maximization program, and expressing (10) as
The
=
allocation induced by the best sustainable
problem that maximizes the ex ante
levels of
Xf
(9)-(12)
program for some sequence
from Proposition
1
as a two-step
F{Kt,Lt) — Ct — Kt+i.
mechanism
utility of the citizens as
therefore a solution to a
is
given in (13), but must also choose
aggregate consumption and labor supply consistent with the sustainability constraint
of the politician in power.
^'''
An
important implication of this result
is
that political
economy
considerations do not fundamentally alter the optimal taxation problem; instea.d, they modify
the aggregate constraints in this dynamic maximization problem.
view, this theorem implies that
we can separate the
From a
analysis of the political
technical point of
economy
of
dynamic
taxation into two parts:
1.
We
first
solve the
problem of providing incentives to individuals
given,
aggregate levels
of consumption and labor suppl}'.
2.
We
then provide incentives to politicians by choosing aggregate variables and the level
of rents.
Accordingly, the best sustainable mechanism will be undistorted
same
"
If,
allocation as that of a full
The key
it
can achieve the
dynamic Mirrlees economy with the same sequence of {xt}'^Q
Theorem 2 is that pohticians' deviation payoffs depend only on aggregates.
F{Kt,Lt), the maximum consumption for the politician were a nonlinear function of the
distribution of labor supplies, [k,i.]ii^j, Theorem 2 would not necessarily hold.
feature necessary for
instead of xi
entire
when
=
23
(which naturally involves no marginal distortions
in
addition to tliose implied by Mirrleesean
optimal taxation).
Another major
Theorem
2 for our purposes
that
enables us to represent the
it
terms of aggregate distortions, corresponding to what the sequences {Ct,Lt}^Q G A°° are
(and
how they
make
further progress along these lines,
from the solution to the dynamic Mirrlees program
differ
we show
entiable in the sequences {Ct^Lf}'^^ e A°°.
{Ct,Lf}5^Q where only one element,
for
is
between the dynamic Mirrlees program and the best sustainable mechanism purely
differences
in
role of
t
7^ s
C.,
We
that U{{Ct,Lt}^Q)
or Ls for
some
specific s
U
mechanism F*
say
th.a,l
We
—
(19) with {Xt]'^Q
{Cf, Lf, /^f+i,X(}^o
{^(}/^o
*^
at
t'
"'^'^
if
^1'
<Ci,Li,Kt+i\
~
'i-s
~
^f' ^f
asymptotically undistorted
^-'t'
by Fl^ and
at time s
dynamic Mirrlees problem where,
is
a solution to
—
^^t'+\
'
(18) subject to
i
say that
H'^e
-^V+i-
undistorted as
it is
if
This definition states that an undistorted allocation
result in the standcird
also denote the partial
the seqv.ence {Ci,Lt,Ki+i,xi}'^Q induced by the best sustainable
undistorted
is
is differ-
varied (with ah Ct, Lt
is
and capital
derivatives of the production function with respect to labor
We
To
with respect to such variations by
Wc,({C't,Lf}J^o) ^"'^ ^L,({C't,Lt}'^o) or simply by Uc, and Uls-
Definition 4
(18)-(19)).
can therefore consider variations in sequences
VVe denote the derivative of
held constant).
A
Appendix
in
in
—*
oo.
exactly the allocation that would
in addition to restriction across agents,
the government also has to finance an exogenously-given sequence of public good expenditures
{Xt}^Q.
If
an allocation
is
undistorted and {Cf,Zt}^o GliitA°°, then
Uc,
Fr^,
= -Ul,
(22)
,
Fk,.,,-Uc,,,=Uc,.at time
at time
t
t
(or as
t
—
>
oo).
Here, the
first
(23)
condition states
given the utility function U{{Ci, L(}^q)
is
tliat
the marginal cost of effort
equal to the increase
additional effort times the marginal utility of additional consumption.
the cost of a decline in the utility by saving one
more unit
in the next period times the marginal utility of
to
in
output from the
The second one
be equal to the increase
consumption then.
Once
requires
in
output
again, these are
aggregate conditions since they are defined in terms of the utility functional W({Ct, L(}f^o))
which represents the ex ante maximal
Moreover,
it is
if
a steady state exists
utility of
an individual subject to incentive constraints.
and the conditions
also clear that {Ci,Li,,Ki+\,xi,}^^^
in (22)
must be undistorted.
and (23) hold as
We
i
^ oo, then
then say that there are no
asymptotic aggi-egate distortions on capital accumulation (or no aggregate capital taxation)
24
.
if
FKt+i
—
oo.
t if
(22)
f
>
=
l^Ct+i
By
is
^Ci and no aggregate distortions on labor supply
if
Uc,
—
Fit
—i^Lt as
implication, an allocation {Ct, Lt- Kt+i, xt}'^Q features labor distortions at time
not satisfied at
side of (22)
t.
We
strictly greater
is
refer to these as
downward
than the right-hand
are intertemporal distortions at time
t,
and
if
side.
labor distortions
If (23) is
the left-hand side of (23)
is
strictly less
Downward
than
distortions
labor supply and less capital accumulation than in an undistorted
less
is
the left-hand
not satisfied, then there
the right-hand side, then there are downward intertemporal distortions.
imply that there
if
allocation.
Best Sustainable Mechanisms
6
We now present
the
main
result of the paper,,
which characterizes the behavior of the sequences
{Cf Lt, Kt]'^Q (and {x/ }^q) and distortions under the best sustainable mechanism. Theorems
,
4 and 6 in the next section provide both further characterization results under additional
assumptions on the persistence of individual types and information structure, and also
clarify
the interpretation of the results in the following theorem.
Theorem
Best Sustainable Mechanism) Consider
3 (Characterization of the
the op-
timal dynamic Mirrlees economy with self-interested politicians described above. Suppose that
Assumptions 1-3 hold and there
exists
{Ct,Lt}'^Q eIntA°° (with Lt
>
Q) for
some
t.
Then, in
the best sustainable mechanism.:
1.
there are
downward
tions at
—
t
1
labor distortions
(provided that
>
t
a.t
some
<
t
oo and downward intertemporal distor-
1).
Let the best sustainable mechan.isni. induce a sequence of consumption, labor supply and capital
levels {C(, Lt, LCt+i}'^^.
Suppose a steady state
{C*,L*,K*), where {C*,L*)
0}, where (/?<!.
2.
if
^—
5,
is
interior.
exists
Moreover,
such that as
let y?
=mf{
t
-^
oc>,
Q G (0,1]
{C(,
;
L/,,
A'j+i}^q —^
p\\mt-^^ q~^Uq^
—
Then:
then there are no asymptotic aggregate distortions on
ca.pital
accumulation and
labor supply;
3.
if If
>
6,
then aggregate distortions on capital accumulation and labor supply do not
disa.ppear even asymptotically
Proof.
is
We show
in
Appendix
a well-defined functional and
A
is
that,
when randomizations
are introduced. U{{Ct, Lj}J^q)
continuous, concave, and differentiable.
suppress randomization to simplify notation.
25
In this proof,
we
We
write the problem of characterizing the best sustainable
Marimon
following Marcet and
mechanism non-recursively
(1998) as
oo
|C(,L,,A,.,:c, ),^0
^^Q
subject to
+ xt +
Kt^i
with ^_i
=
Ct
for all
t,
where
constraint (12).
we
^,
=
+
^,;_j
The
differentiability of
He,
Since
>
/i,
>
and
/j.(_i
Part
is
and ^V't >
W({Ct,
LJ^q)
1:
-
ji^j^^
=
t.
-
[Wc,+,
>
/./.(,
-
S'ifi^^i
implies that for
{Ct,Lt}^o eIntA°°,
i.e.,
whenever 0^ >
for all
But
t.
/if
cian (12)
=
must bind
distortions at that
i-
at
t,
t
>
t
if
0-
Then, no consumption
0.
>
Lj
for
any
then the
/,,
t
with
i,hf
>
0.
Then
is
feasible
and the associated
than Lt
utility to citizens
(26) implies that there will
(27) implies that there will be
=
Thus we must have Lt
t.
=
Ct
—
cannot be optimal. Therefore, the sustainability constraint of
some
and
(27)
0,
the
politi-
be downward labor
downward intertemporal
distortions
1,
We
start
strictly less
than
Part
and
for all
by hypothesis, {Ct,Lt}flQ eIntA°° with Lj >
for all
(26)
>
'tl't+i
in this case,
{Ct,Lt}'^Q eIntA°^ necessarily gives higher ex ante
plan with Lj
=
and
improve by expropriating the entire output at
Since,
Fi,
labor and intertemporal distortions whenever
to obtain a contradiction that
—
•
fit)v'(F{Kt+i,Lt+,))] Fk,,,,
downward
respectively,
allocated to the politician, xt
for all
2:
by proving that
1.
To
{Ct,L(, A'(+i}^Q -^ (C*, L*,
ip
=inf{ g G
/\'*),
thus {C(}^g
all
bounded
is in
:
plimf_oo Q~*'Uq^
for the constraint (16),
it
lies in
the space
c of
—
0}
is
well-defined
convergent infinite sequences
infinite sequences, t^).
the space of sequences {t/zlj^o such that X!*^o
multipher
[0, 1]
see this, recall that by hypothesis, a steady state exists, so that
(rather than simply in the space of
is,
the Lagrange multiplier on the
- i2t^,)v'iFiKt.^))Fu = -Wc,
5'(^,
there will be
M/.-i'
Suppose
politician can
at
is
(25)
have:
Ul,
/j.(
V'^
< F{Ku Lt), and
l^'i
^
°°- Since Uc,
The dual
is
Rearranging equations (26) and (27) and substituting
for
=
/xy(x*)
26
hence
</?
<
e.g.,
that
Luenberger,
1.
Uc, and taking the limit as
we have
WcFt,{K\L*)
0,
is £i,
equal to the Lagrange
the dual space of {C/,}^g (see,
1969, Chapter 9), thus in £i, wliich implies that lim(_,ooWc,
of c
t
—
»
oo,
and
Fk„AK\L*)Wc^^^
where
By
with respect to Xf then implies:
first-order condition
construction,
go to
^u,
is
an increasing sequence, so
Since as
infinity.
and Uq
i
—
proportional to
is
=
5,
we have that
(where the fact that ^*
Therefore, {^^
Part
t
^
3:
—
Mt^i)//-*t
-
follows
0'
*
Suppose that f >
This implies that
oo.
<
-
Mt
'*
/^-t+i
^''+'
= Li
..
5'+\^'{x*)
(/x,
from Part
t --»
1,
since
first
/i*
or
~
Pt|
>
/./,,_|_i
cx).
(31)
^0 and
and
fl^
/ij
-+
/Xj
>
/x*
for
G
(0, oo]
some
t).
6.
—
In this case, (30) implies that
^.(_j,)//Z{
>
as
—+
t
oo, so
Uq
is
proportional to
from (26) and
(27),
(/?*
as
aggregate
a
part of the theorem states that the sustainability constraint of the politician,
arises because, as
Intuitively, this additional (aggregate) distortion
output increases, the sustainability constraint (12) requires that more rents
be given to the politicians
The best
in
These additional rents increase the
power.
sustainable
mechanism
undistorted allocation, a small reduction
but a first-order decline
in the
in
efl:ective cost
of
creates distortions so as to reduce the level of
output and thus the rents that have to be paid to the
in output,
some value
and distortions disappear asymptotically.
(12), necessarily introduces a distortion.
production.
^
t
oo, j^t+i
distortions cannot disappear, completing the proof,
The
as
,,
(31) implies that as
>
—
either converge to
(30) can be written as
^9*,
^
^p
must
it
oo an interior steady state {C*\L* ,K* ,x*) exists by hypothesis
>
5'v'{x*)
Since
^
derivatives are evaluated at the hmit {C*,L*,K*).
all
The
{^,,^,-„^)v'{F{K\L*))FK,,^,{K\L*)
_^
politician. Intuitively, starting
from an
labor supply and capital causes a second-order loss
amount
of rents that
Consequently, some amount of distortion reducing
Icibor
need to be paid to the politician.
supply and capital
is
optimal from
the viewpoint of the citizens.
Part 2 states that as long as an interior steady state exists and
rapidly (which
is
will
be
declines sufficiently
related to the rate of discounting by the citizens, see below), the multiplier of
the sustainability constraint goes to zero. This result
run there
Uq
is
important as
"efficient" provision of rents to politicians,
it
implies that in the long
with the necessary tax revenues
raised without distortions. Intuitively, current incentives to the politician are provided by both
consumption
in the current period, Xt,
and by consumption
in the future.
by the politician not only relaxes the sustainability constraint
27
Future consumption
in the future
but does so in
all
Thus,
prior periods as well.
all else
equal, optimal incentives for the politician should be
Backloading leads to sustainability constraint not binding
backloaded.
run and
in the long
distortions disappearing.
Notice that the results in this theorem compare 5 to
ante marginal utility of consumption W^^
where W({G,i'(}^o)
We
will
show
it
time
i+
next section that in two important cases, this
in the
We
show
will
^p is
the rate at which the ex
declining in the steady state. Clearly, in the case
is
^p is
different
which the
Part
from
politician
3,
ciently low
is
Therefore, the case oi
<p
—
in general
it
indeed corresponds to a situation in
5
if
the discount factor of the politician 5
to the fundamental discount factor
disappear, even asymptotically.
The
(^,
realistic political
significance of this result
economy models
citizens, this part of the
politicians are
theorem implies that
in a
is
suffi-
then aggregate distortions will not
aggregate capital taxes in contrast to the existing literature on
many
though
/3,
e.g.,
as patient as the citizens.^''
on the other hand, states that
compared
/3.
compares with one unit of resources at
t
next section that without any dynamic incentive linkages,
in the
B.
exactly equal to
indeed be the case. In
will
with constant types, this fundamental discount factor coincides with
may be
is
the fundamental discount factor of the citizens,
measures how one unit of resources at time
1.
Here
time separable, the rate at which W^, declines
i^
the more general case of present section,
since
^p.
is
that
it
dynamic
also implies positive
Since in
fiscal policy.
—or act as — more short-sighted than the
number
of important cases, political
economy
considerations will lead to additional distortions that will not disappear even asymptotically.
7
Constant Types or Private Histories
Theorem
3 provided a complete characterization of the distortions introduced by political econ-
omy and commitment problems and
the results of
for all
i
and
Theorem
t,
that
is,
6.
We
first
economies
remain constant thereafter. This
their asymptotic behavior. In this section,
focus on economies with constant types, where 9\
factor
(/J
is
is
the assumption that
(e.g.,
With constant
is
used in
histories,
this case allocations
where individual
much
^J^j
and
of the literature
on
Roberts, 1984, Freixas, Guesnerie and Tirole,
types,
we show that the fundamental discount
identical to the discount factor of the individuals,
with private
=
in whicli individual types are realized in the first date
dj'namic mechanisms without commitment
1985, Bisin and Rampini, 2005).
we strengthen
histories will not
j3.
We
next turn to economies
be observed by the politicians (so in
can only be conditioned on current reports). This restriction enables us
^^In part 2 of this theorem, we limit attention to the case in which ^ = 5, since when i/j < 5 we will not
converge to an interior steady state. Theorem 5 in the next section explicitly deals with non-interior steady
states.
28
main
to focus on the
interplaj'
substantial complications
this case,
we
between private and public incentives, without introducing the
when
tliat arise
show not only that
will
^—
distortions will be present starting at
individuals are given complex
dynamic
economy and commitment
0, but also that political
and that the characterization
t --
incentives. In
results hold without
assuming that the best sustainable mechanism induces an allocation converging to a steady
state.
Best Sustainable Mechanism with Constant Types
7.1
Since individual types do not change, truthful reporting
Let us start with constant types.
along the eciuilibrium path
Theorem
(cf.
1)
implies that individual incentive compatibility
e)
> Y^p'u{ci{e)MO)
constraints can be simply written as
oo
oo
f3'uic,i9)Jt{9)
for all 9
Q
^
Lt and Cj,
power,
e Q. Since types are known at
and
vary in this case
will
In this case,
xi.
Theorem
I
is
all
O)
(32)
I
dates, the only reason
result.
Mechanisms with Constant Types)
Consider the op-
timal dynamic Miriiees economy with self-interested politicians described above.
types are constant, thai
is,
6]
—
Assumptions 1-3 hold and these
the best sustainable
1.
downward
tions at
—
1
6]^i for all
exists
i
£ / and
t
=
>
that
Suppose moreover that
0,1,...,
{C(,Lf}J^o G/niA°° (with Lt
Assume
0) for
some
t.
Then, in
mechanism,
there are
t
aggregates,
because of changes in the rents paid to the politician in
we have the following
4 (Best Sustainable
why
labor distortions at
(provided that
t
>
some
t
<
oo and downward intertemporal distor-
I).
Let the best sustainable mechanism, induce a sequence of consumption, labor supply and capital
levels {Cf, Lt,
/\(+i}^q. Suppose a steady state exists such that ast—> oo, {Ct, Lt, Kt+\}t—o ~*
{C*,L*,K*), where {C*,L*)
2.
if j3
—
5,
is interior.
Then:
then there are no asymptotic aggregate distortions on capital accumulation and
labor supply;
3.
if
p >
5,
then aggregate distortions on capital accumulation and labor supply do not
disappear even asymptotically.
Proof.
Sint:e lliis
proof deals explicitly with randomizations,
A.
29
it
is
provided
in
Appendix
Best Sustainable Mechanism with Private Histories
7.2
We
In this subsection, we focus on private histories.
notation by assuming that within each period, there
types, denoted
of the
by G, and
economy
also
=
and the
an aggregate invariant distribution of
by removing capital, so that the aggregate production function
is
Yt
where Kq
is
also simplify the exposition
= Lu
(33)
and Lt denotes the aggregate labor supply
without any significant consequences
Private histories imply that
agents' current report.
in
at time
t.
These simplifications are
our analysis.
for
admissible mechanisms, allocations must depend only on
In such an environment the incentive compatibility constraints for
agents can be separated across time periods, and written as
u
e
for all Of
The
6
and
6t,
G 0, and
best sustainable
(12), (34)
{ct {6,)
,
/,
{61)
for all
\0,)>u
(c,
(Ot^
,
/,
mechanism with private
et)
histories therefore
(34)
maximizes
(9)
subject to
and the resource constraint
+ xt<Lt.
(35)
Returning to the quasi-Mirrlees program defined above,
pri\'ate histories, the
in the
I
t.
Ct
with
(^,)
optimal allocations of
same period, Ct and
that U{{Ct, LtYZo)
is
Lt,
immediately imply that
The program
and are independent
time separable,
valued differentiable function
{ct, It)
U
U (C, L)
R^
:
is
i.e.,
The
straightforward to see that
of any Cs, Ls with s
^
results for
U{{Ct,Lt}'^Q)
and
This implies
t.
= E YlZo l3^U{Ct,Lt)
also well defined, concave,
for the best sustainable
is
depend only on the aggregate variables
U{{Ct, Lt}^o)
-^ M.
it
for
in
some
real-
Appendix
A
differentiable.
mechanism, (20)-(21), can now be written
as:
00
max
>
/?'[/(a,L,)
(36)
subject to the resource constraint, (35), and the sustainability constraint.
wi.
=E
Y^5-'v{xt+,)
>
viLt),
(37)
.s=0
for all
t,
where wt denotes the present value of
utilities delivered to
30
the politician at time
t.
.
The
incentive compatibility constraints for individuals in (34) play a similar role to (14) in
our formulation above. In particular, we can define a static set of feasible aggregate consumption
and labor supply
A=
levels,
(C, L) such that 3 {c [9) ,l{e)] satisfying (34),
{
c=
Finally,
we
also
I
c{e)dG[e),3.iidL=
and
(38)
i(e)dG{9)}.
I
adopt the following sustainability assumption, which
lishing convergence to a steady state
and
in part 2 of the next
be used in estab-
will
theorem
when
(in particular,
the utility provided to a politician reaches the boundary of the set of feasible values).
w=
Assumption 4
Intuitively, this
the politician
~
S).
-5) >
/ {1
e argmax(c,L)eAi' (L
— C)
The concept
/ (1
—
(5),
V (L)
assumption ensures that the highest discounted utihty that can be given to
sufficient to satisfy the sustainability constraint (37). Clearly this
is
when
satisfied
/ {I
(sustainability) There exists {C,L)
v{L~C)
such that
is
- C)
max(c,i)eA v {L
Let
the politician's discount factor,
of the aggregate distortion
is
assumption
6, is sufficiently large.
When
also simpler in this setup.
the solution to the dynamic (full-commitment) Mirrlees program (18)-(19)
GlntA
(C, L)
satisfies:
Uc{C,L)^--ULiC,L),
where Uc and Ul are the
partial derivatives of
a downward labor distortions
if
U
(39)
(C, L) with respect to
the left-hand side of (39)
is
C
strictly greater
and L.
We
refer to
than the right-hand
side.
The main
Theorem
result of this section
5 (Best Sustainable
economy with no
capital
is
the following theorem:
Mechanisms with Private Histories)
and with private
histories.
Consider the
Suppose also that Assmnptions
1,
3,
and
4 hold.
=
1.
At
2.
Suppose that P <
t
0,
there is
values {uit}'^Q.
an aggregate
distortion.
Let T* be the best sustainable
5.
Then
{u'<}'^o ^^
'^
mechanism inducing
a sequence of
non- decreasing sequence in the sense that
ti'^+i
>
wt
for allt. Moreover, a steady state exists in that {u'(}^q converges (almost surely) to some
w* G
[0,
u.']
and {C/,
L(, .tj}J^q converges (almost surely) to
asymptotically undistorted.
31
some {C* L* ,x*), which
,
is
.
If
3.
P >
5,
then aggregate distortions do not disappear even asymptotically.
Proof. Most of the
in
i
Theorem
=
1
5
and
theorem follow as
corollaries of the corresponding results
three additional results are that there are distortions at the initial date,
rather than at some possible future date, that {tt'(}J^o
0,
when
The
3.
results in this
<
/3
2 in
a steady state necessarily exists,
^^
and that
^ non-decreasing,
All three of these results follow
from Theorem
Acemoglu, Golosov, and Tsyvinski (2007a), and we do not repeat these proofs to
economize on space.
Theorem
5 provides a tighter characterization of the best sustainable equilibrium for the
important special cases of private histories than the general results
in
Theorem
3.
It also
enables us to see the role of the relative discount factors of the politicians and the citizens
more
clearly.
More
Part
specifically,
some period
rather at
>
<
1
Theorem
of
0.
It is
5 establishes that there
possible to
is
distortion in period 0,
compare the discount
factor of the politician
5 to the discount factor of the agent as function
U{C,L)
that a sequence of values delivered to politicians, {wi}'^q
is
is
w
is
reached the allocation
for the case of politicians being
Finally, Part 2 of the
more patient than
if
also does not
the boundary
Theorem extends
results
agents.
Interpi^etation of Distortions
7.3
We
The theorem
Assumption 4 guarantees that
be undistorted.
will
show
non-decreasing providing an easily
interpretable notion of backloading of incentives for politicians.
require existence of the interior steady state.
We
separable across time.
can also use the economies with private histories or with constant types to
meaning
of aggregate distortions.
economy with private
To do
this in the cleanest possible fashion, let us focus
histories (the results are identical with constant types)
following additional assumption, which
clarify the
is
typically
imposed
on the
and introduce the
analyses of static and dynamic
in
incentive problems:
Assumption
Then Uc
(c,
Given
I
\
5
(single crossing) Let the partial derivatives of u he denoted by Uc and
0) / \ui (c,
/
|
^)|
is
increasing in 6 for all c and
I
and
all 9
ui
£ Q.
this single crossing property, the set of incei:itive coi^rpatibility constraints
with
pri-
vate histories, (34), can be reduced to a set of incentive compatibility constraints only for
neighboring types. Since there are
N+1
types in 9, this implies that (34)
is
equivalent to
A''
incentive compatibility constraints. This then enables us to establish the following proposition,
which
illustrates the relationship
between aggregate distortions and individual income taxes:
32
Proposition 2 Suppose Assumptions
and 5 hold and suppose that the
1
best sustainable
mech-
anism does not involve randomization. Consider a sequence of {Ct,Lt]'^Q. Then:
the marginal labor tax rate
1.
l
2.
if
{Ct,
^t}^o
JS
undistorted at
i.e..
u^
Proof. Assumption
[ct
[On)
Since there
where Xni
is
at time
vct
t,
time
t
is
given by TN,t
~
and
the labor supply decision of the highest type of agent is
k [On)
ui
\
9n)
-'"/, {ct
we only need
{0^)
,
k {On)
\
On)-
to check incentive compatibility constraints
be the partial derivatives of u (which exist by Assumption
= yCU
Uc{Ct[ON)Jl[ON)\ON){l
+
n
+ ^m) -
(ct
[On)
,
k {On)
\
On)
(1
XNt)
-i^Lt,
the multiplier on incentive compatibility constraint between types
is
the multiplier on (16) at
U {C,L)
and the
t
first
{ct
[On)
,
k {On)
\
On)
i//,f
the multiplier on (17) at
is
Lagrange multipliers,
I'ct
— Uc
_,
~
UiACtJUl
Uc {Ct,Lt)
'"''''
^
result follows
t.
By
the
{Ct,Lt) and
'
equality defines tn.i, and the second equality establishes the
lemma. The second
9n and 0n-i
we have
ecjuations,
Ui{Ct{0N),lt{0N)\0N)
Uc
and
definition of
—UL{Ct;Lt)- Combining these
where the
^
no randomization, we have
is
differentiability of
=
,
t,
5 implies tliat
for neighboring types. Let Uc
i^Lt
at
+ UL{CuLt)/Uc{CuLt).
undistorted,
1).
on the highest type of agent, 9^,
immediately from setting UL{Ct,Lt)
first
part of the
— —Uc{Ct,Lt)
from
the definition of an undistorted sequence, in particular, equation (39).
This proposition therefore further
clarifies
the meaning of the aggregate distortions, which
have been our focus so far and shows that they are naturally linked to the marginal labor
taxes in the standard Mirrlees problem. In particular,
if
in
the standard Mirrlees problem, the
marginal income tax on the highest type should be equal to zero, then the added distortion
is
exactly equal to the tax that will be imposed on the highest type under the best sustainable
mechanism.
8
Example
We now
briefly illustrate the results of
Theorems
3
and 4 using a simple example and then
provides some illustrative computations that show the dynamics of distortions and allocations
under
different values of the discount factors for the citizens
33
and the
politician.
Suppose that
6=
there are constant types, drawn from the set
{Oo,0i} and the utility function
is
u{cJ\e) = uic)-xil/0),
where u
continuously differentiable, increasing and strictly concave and x
is
differentiable, increasing
and
Furthermore, suppose that u
strictly convex.
TT
an individual
type.
To
is
Inada-type
take 9o
—
0,
so
again disabled and cannot supply any labor. Suppose that with probability
born, and remains, as high type, while with probability
is
continuously
satisfies
We
conditions, so that first-order conditions are always satisfied as equality.
that the low type
is
1
-
tt,
he
a low
is
simplify discussion, let us again ignore randomizations across the consumption and
labor supply levels of different types. Then, the quasi-Mirrlees problem can be written as
oo
{c,(Oo),cii9i)MSi)}Zo
^
jr^
(40)
subject to
TTCf
+
(^i)
(1
-
tt)
q
<
(Oq)
i\li
{0\)
—
xt for all
oo
^
=
irlt
(Oi)
and
oo
\a {ct (01 ))
/3'
-X
> 2] /3*
ih iOi) /Oi)]
[^ (ct (^o))]
,
t=0
1^=0
where Lt
t,
and Ct
—
while the second constraint
is
Lt
—
The
xt-
first
constraint
is
the resource constraint for each
i,
the incentive compatibility constraint sufficient for the high type
to reveal his identity given the presence of effective
Assigning Lagrange multipliers
and
/3'/i(
commitment along the equilibrium
path.
respectively, to these constraints, the first-order
//,
necessary conditions of this problem can be written
as;
77)u'(ct(0i))
=
TT/it
(41)
(l-^-r;)u'(ct(0o))
^
(l-7r)M„
(42)
+
(7r
and
^^^"'^\'(M^i)/^i)
=
(43)
M.^-
Ox
Equations (41)-(42) imply that
u'{ct{ei))
u'(c/.(0o))
Consequently, there
if
a steady state
Ct {Oq)
—
»
c°*,
and
is
constant risk-sharing between the two types in
exists,
It
so that Xi
[9i) —* /*,
in the previous section.
^ (l-7r-7/)
(^ + '?)
Put
—
»
and hence
x*
p.,
(41)-(43)
,
—
>
/n*.
all
Moreover,
combined imply that ct{9i)
Consequently', in this case
differently, in this particular case, the rate at
34
periods.
(/?
=
^
—
*
c^*,
as claimed
which the derivative
0.15
-
01
0.05
Figure
1:
The time
-
patli of distortions for constant types witli
Higher curves correspond to lower values of
Uq
declines
citizens,
i.e.,
We now
easy to determine, and
is
(/?
=
it
/3
=
0.9
and
5
—
0.6, 0.7, 0.8, 0.9.
5.
does so at
same
tlae
rate as
tlie
discount factor of the
/?.
illustrate the results presented in the last
two sections numerically. Suppose that
individual utility functions take the form
u{c,l\e)^
Suppose
^1
=
tiiat
type Oq
is
it
=
1/2 of the population
utility function of tire politician
is
given by v [x)
so that the production function
is
F {K, L) =
We
case,
0.7,
show the aggregate
with
and
<5
=
0.9,
and
(44)
disabled and cannot suppl}' anj' labor, so ^o
Let us also assume that a fraction
1.
^- —.
distortion, 1
=
-s/x.
We
is
=
0,
and we normalize
of type 9i
and
tliat
the
consider the case without capital,
L.
+UiJUci-,
in
Figure
1
for politician for the baseline
also for a range of lower discount factors for the politician, 5
=
0.8,
0.6.
Consistent with part 2 of
Theorem
3,
when
aggregate distortion converges to zero and
resulting from political
tire
/?
=
5
=
0.9,
convergence
is
the lowest curve shows that the
rather fast
economy and commitment disappear very
—so that distortions
quickly. Instead, wlren 6
<
/3,
the aggregate distortion converges to a positive, and potentially large, asymptotic value. For
example, when 5
=
0.6, the
aggregate distortion converges to an asymptotic value of 0.15 (the
highest curve in the graph).
Another question concerns how much of the economy's output has to be allocated to
politician (as rents or
government consumption).
35
tlie
Figure 2 answers this question, again for
/3
=
0.9
the citizens,
As we
=
and 5
0.9, 0.8, 0.7,
and
When
0.6.
the politician's discount factor
equal to that of
receives a very small fraction of the output even in the asymptotic equilibrium.
it
consider lower discount factors for the politician,
consequently,
is
it
its
temptation to deviate increases and
receives a higher fraction of the output.
r
r
Figure
2:
Time path
to lower values of
of XtjYt with
/3
=
0.9
and 8
—
0.6,0.7,0.8,0.9. Higher curves correspond
d.
9
Benevolent Time-Inconsistent Governments
The
analysis so far has focused on the case
though
this case
is
of relevance for
many
when
political
politicians were purely self-interested.
economy
applications,
also
it is
Al-
important to
understand how the results generalize to the case considered by Roberts (1984), Freixas, Guesnerie
and Tirole (1985), or Bisin and Rampini (2005), where the government
but "time inconsistent",
now
i.e.,
unable to commit to a
full
is still
benevolent,
dynamic mechanism. To do
this,
we
consider a more general utility function for the government of the form:
oo
E
b'
(1
-
a) V [xi^s)
+
«
Et+,
/
u
{ct+s.
k+s
0'+')
t+s (ai+s^
[9
dG'+'
I
(45)
.s=0
where the second term
<
a
<
1."^
is
the average (expected) utility of the citizens at time
Therefore, this utility function
government when a
=
is
i
-f-
s,
and
identical to that of a purely-self-interested
0.
^''We do not allow the case of a
Our environment obviously
=
1
(purely benevolent government) as
it
requires a different proof strategy.
allows the government to be arbitrarily benevolent (a
36
^
1
and
S
=
0)
)
To incorporate the
change the
political
government out of
of the
main
case of a benevolent time-inconsistent government,
game.
office.
we no longer allow
In particular,
Instead, the
same government
we
also
citizens to vote the current
always in power. Despite
is
results so far continue to hold, since citizens have another effective
against the government, to produce zero.
In particular,
need to
this, all
punishment
the government deviates from the
if
prescribed policy (or from the implicitly-agreed social plan), individuals can exercise their
freedom on labor supply and produce zero output thereafter. ^^
game form
the results so far hold under this alternative
The advantage
2006).
game form
of this alternative
is
can be verified that
It
(e.g.,
utility,
which
is
of
Acemoglu, Golosov and Tsyvinski,
(see
that
naturally adapts to the case of
it
a partly benevolent government. In addition, we also need to strengthen Assumption
assume separable
all
and
1
a standard assumption in most analyses of dynamic taxation
Golosov, Kocherlakota and Tsyvinski, 2003, Kocherlakota, 2005).
Assumption
1'
(separable utility) u{c,l
—
9)
\
u{c)
—
\{l
continuously diff'erentiable, strictly increasing and concave, and x(ferentiable, strictly increasing
and convex for
and
9 £ 0,
all
where u
0),
\
|
is
9)
satisfies
X
(0
:
K+
-^
M. is
continuously
|
—
6*)
dif-
for
all
0ee.
The next theorem shows that Theorem
1
and Proposition
1
continue to hold in this more
general environment.
Theorem
ment
6
utility IS
(Truthful Revelation with Benevolent Government) Suppose that governgiven by (45) and that Assumptions
of strategy profiles
F and a
1
',
2 and 3 hold.
mechanism, there
that support a sustainable
of equilibrium strategy profides T* and. a*
=
(a*
|
Then for any combination
a') for
some
a' such that T* induces direct
submechanisms a* induces
truth telling along the equilibrium pa.th,
^r, a] =/[r*, a*] and x
=
,
[F, a]
x [T*,a*]. Moreover, the
tion to maximizing (9) subject to (10), (11)
and
the
another pair
exists
and c[T,a]
best sustainable
—
mechanism
c[T*,a*],
is
a solu-
government sustainability constraint:
oo
Y;,5'[{l-a)v{xt+s)
a (]E,+,
I
+
[u (c {9'+^))
-x{l
max
x[+f
for
all
c',
{0')dG<
(0'
(^'+1
{l-a)v
Ot+s)]
dG'+^ {9'+^)
+
u
>
I
[xt)
a
(c, (9*))
dG'
{9')
,
(46)
J
<F{Ki. ,Lt)
t.
^'With these punishments
strategies, eqiiiUbria are no longer renegotiation-proof. Nevertheless, it is possible
extend the game considered here, so that even though politieians are partially benevolent, there is still
replacement of politicians (and a politician who is replaced still cares about the average utility of the citizens),
and obtain similar results. We do not introduce this somewhat more involved game form to economize of space.
to
37
Proof. See Appendix B.
The
difference in the proof with the previous environment
now
pohticians,
its
that instead of replacing
agents use the nuh strategy following the deviation by a politician.
imagine that the government has undertaken a deviation in which
ticular
of
is
has used some
it
past information in order to improve the ex post allocation of resources.
clearly be desirable given the utility function of the
with the Roberts' (1984) example,
it
may have
government
In par-
This could
in (45), but as illustrated
very negative consequences ex ante. Therefore,
the best sustainable mechanism will have to discourage such deviations. To do this, imagine
a punishment strategy, in which following any type of deviation,
To
labor.
establish
Theorem
When
sequentially rational.
all
we need
all
6,
to
assumed
among
redist ribute the rest equally
in
Assumption
Since there
1'.
is
that such punishment strategies are
other agents choose zero labor supply, following any deviation
to positive labor supply, the government would
and would
show
individuals supply zero
all
consume some
all
of the increase in output
itself,
agents given the separable utility function
a very large number of citizens, this implies the
is
deviating individual will receive no additional consumption from supplying positive labor, and
thus
it is
sequentially rational for
all
by the
citizens to supply zero labor following a deviation
government.
This theorem therefore shows that revelation principle applies to the case of benevolent, but
time-inconsistent governments as well, though under the additional assumption of Assumption
1'.
The next example shows why
Example
To avoid
1
in
6 6 {0, 1}, with 6
utility u{c,-
\
9
=
where with \i ()
in labor
supposed
9
—
I
for this
to
Now
=
>
among continuum
example, where n
of agents, let us consider a finite
large (exactly the
is
who can
corresponding to the disabled type,
=
0)
u{c), while the utility of type 6
strictly increasing in
same example can
is
I.
=
1 is
only supply
u{cj
\
=
1)
I
=
=
0,
u{c - Xi{l)),
Furthermore, suppose that aggregate output
fully benevolent,
i.e.,
a
=
1 in
terms of the
and has
is
linear
utility function
imagine the economy has entered the punishment phase where each citizen
to supply
I'
necessary:
is
an economy with a continuum of agents). There are two types of agents,
and that the government
in (45).
assumption
issues of deviation
economy with n agents
be constructed
this
I
=
and consume
such that \i
{I')
distribute consumption (output
/'
<
>
1.
—
c
0.
Consider a deviation by an agent,
0) to
maximize
its
own
utility,
which involves maximizing
utility of
consumption across agents,
i.e.,
(c,)
=
u'
of type
Following this deviation, the benevolent planner will
average utility of the citizens, thus equating the marginal
u
i',
is
[ci'
—
\i
38
(/'))
for all
i ji i'
thus,
Ci^
—
Ci'
-
{I'
c,:
+
(l'))
Xi
Xi
for all
(/')
/n and
c^>
=
^
i
[l'
-
The
i'.
Xi
{I'))
/n
resource constraint
+ X\
(^')-
The
is
{n
-
l)c,;
+
=
d'
I',
resulting utility of individual
i'
or
is
u{{l'-Xi{l'))/n)>u{G),
any
for
n,
thus giving him greater utility than supplying zero labor.
punishment phase where each
citizen
is
This proves that the
supposed to supply zero labor
is
not sequentially
rational and thus cannot be part of a (Perfect Bayesian) equilibrium with this utihty function.
The next theorem provides
ymptotic behavior
and the
a characterization of the structure of distortions and their as-
citizens have the
same discount
since constant types constitute the
main
and under the assumption that the
for the case of constants types
results in a succinct fashion.
factor,
i.e.,
p =
6.
We
politician
start with this environment,
most commonly-studied case and enable us to obtain the
Theorem
8 below generalizes the results of this theorem to
non-constant types.
Theorem
stant Types) Suppose that government
sumptions
au' (0)
and
7^ (1
Government and Con-
7 (Best Sustainable Aiechanism with Benevolent
1
—
a)
'(/
(0),
given by (45) with a G
(0, 1)
and
Furthermore, assume thai there are constant types,
2 and 3 hold.
',
utility is
fi
that As-
=
5
and
Then, asymptotically there are no aggregate distortions on labor supply
capital accumulation.
Proof. See Appendix B. H
This theorem implies that in an economy with constant types, aggregate distortions
appear regardless of the degree of benevolence of the government.
Consequently, there will
be no aggregate capital taxes and no further taxes on labor bej'ond those implied by the
commitment
Mirrlees economy."" In the case where a
to the fully-benevolent case,
where
Once
in a very similar
again, the
main source
of the difference
the information
it
is
arbitrarily close
results in Roberts (1984),
is
the infinite-horizon nature of our economy,
which the government
will
be punished
if it
exploits
gathers via the earlier submechanisms.
this section
general result.
the government
full-
environment, the equilibrium always involved extreme distortions.
to construct ecjuilibria in
end
1,
and the theorem contrasts with the
which allows us
We
^
dis-
and the paper with a generalization
Theorem
3.
of
Theorem
7,
which parallels our
This theorem illustrates the importance of the discount factor of
the politicians in the environment with that partially benevolent government, though, now, the
fundamental discount factor to which the discount factor
of the politicians
is
being compared
^^The assumption that au (0) ^ (1 — a) i/ (0) rules out a special case in which our method
work (though other more complicated approaches may worlv even without this assimiption).
39
of proof docs not
to
is
a moie complicated object than
Theorem
defined in
'^
3,
making the
result
somewhat
wealcer.
For this theorem,
which now has
Theorem
now
{mj It^o denote the multipliers associated with the sustainabihty con-
government and
straints of the
in
let
let
on
to be conditioned
^ =inf{
3, let
q
e
a function of the sequence
Theorem
U ({C(, Lt}^o;
Case) Suppose
that
government
:
1.
is
utility is
given by (45) with a £ (0,1), that Assumptions
>
Q) for
some
t.
1',
Then, in the
mechanism:
there are
downward
tions at
—
t
0} (which
Mechanism with Benevolent Government: General
2 and 3 hold, and that there exists {Ct, Lt]'^Q eIntA°° (with Lt
best sustainable
^
phm,._oo ^"'t/^, ({Q, iil^oi (m* }So)
as well as {Q*, L,*}).
{/j,*}
(Best Sustainable
8
sequence of multipliers as well (see Appendix B). As
this
[0,1]
{lAYtLf^) denote the indirect utility functional,
1
labor distortions at
(provided that
>
t
some
t
<
oo and downward intertemporal distor-
1).
Let the best sustainable mechanism induce a sequence of consumption, labor supply and capital
levels {Ct, Lf,
Kt+i}^Q. Suppose a steady
{C*,L*,K'*), where {C*,L*)
2.
if
^=
5,
is
interior
state exists such that
ast^
oo, {Ct, Lt,
Kt+i}'^Q -^
Then:
then there are no asymptotic aggregate distortions on capital accumulation and
labor supply;
3.
if If
>
5,
then aggregate distortions on capital accumulation and labor supply do not
disappear even asymptotically.
Proof. See Appendix B.
This theorem therefore shows that the fundamental results from Theorem 3 generalize to
the case with a partially benevolent government.
is
now an even more complicated
this
this case
when types
in
Theorem
3.
Nevertheless,
Theorem
are constant, sharper results can be obtained.
results are exactly those that are necessary for rethinking the efficiency of
mechanisms
10
in the presence of
if,
object and certainly not easy to compute, the results in
theorem are weaker than those
even in
Since the fundamental discount factor,
commitment problems on the
7
showed that
Moreover, these
dynamic tax-transfer
side of the government.
Conclusions
In this paper,
we took a
ear taxation. Political
first
step towards a political-economic analysis of
economy considerations become
40
dynamic and nonlin-
particularly important in the context of
nonlinear taxation, which involves a significant
amount
and enforcement power
of information
being concentrated in the hands of the government and policymakers. Unless these policymakers are benevolent
and able to make commitments, the structm'e of taxation must not only
provide incentives to individuals, but also respect political economy constraints
—that
is,
it
should provide the appropriate incentives to policymakers. Despite the complex set of issues
that arise in balancing private and public incentives,
tractable framework for the analysis of
how
political
possible to develop a relatively a
is
it
economy constraints
affect the structure
of taxation.
To
achieve this objective,
we focused on the best sustainable equilibrium,
equilibrium that satisfies the incentive compatibility constraints of politicians.
how
sustainable mechanisms, where the politician in power
is
is
We
showed
given incentives not to misuse
resources and information, can be constructed in the infinite-horizon
important result of our analysis
the best
i.e.,
economy we
study.
An
the revelation principle along the equilibrium, path, which
shows that truth-telling mechanisms can be used despite the commitment problems and the
different interests of the
government
(politicians)
and the
Using
citizens.
we
this tool,
pro-
vided a characterization of the best sustainable mechanism. Political economy considerations
introduce additional constraints on the optimal taxation problem, but these constraints are
intuitive
and
relatively simple to characterize. In particular,
we showed that the provision
incentives to politicians can be separated from the provision of incentives
agents. Political
economy
economy) distortions and change the structure
Our main
time.
We
of taxation.
results provide a characterization of these distortions
showed that when
politicians are as patient as, or
and their evolution over
more patient than,
The
therefore implies that the insights from Mirrlees' classical analysis
constraints and
literature
may
also
and from the more recent
show that when
economy
politicians are less
patient than the citizens, aggregate distortions remain positive even asymptotically.
case, in contrast to the classical results in optimal taxation, there will
and positive aggregate
capital taxes even in the long run.
power
This result
generalize to certain environments featuring political
commitment problems. However, we
citizens,
politician in
receives rents, but these rents are provided without additional distortions.
dynamic taxation
on ag-
These constraints then lead to new
aggregate capital and labor distortions disappear in the long run.
still
and insurance to
constraints, instead, take the form of additional constraints
gregate consumption and labor supply in the economy.
(political
of
In this
be positive distortions
To the extent that smaller discount
factors for politicians than for the citizens are a reasonable approximation to reality, our results
also suggest a possible explanation for understanding distortionary long run taxes
capital.
41
on labor and
Our
infinite
analysis relied on the infinite horizon nature of the
economy and
especially
on the
planning horizon of the politicians. Nevertheless, similar can be developed even when
politicians have finite horizons,
and a detailed investigation
of this issue
would be an interesting
area for future research. For example, we conjecture that in a model with either finitely-lived
politicians or with
term
limits, a society consisting of infinitely-hved citizens or
generations of citizens will be able to commit to providing a continuation value
to politicians that have not deviated from the social plan. In this case, even
will
(e.g.,
overlapping
"pension")
though distortions
not disappear in the long run, they should decline during the tenure of the politician.
Such a model would
also enable
an analysis of the
institutional constraints on politicians.
42
effects of
term
limits
and other
realistic
Appendix A: Properties
Some
We
U {{Ct, LiYiIq)
of
Technical Results
first
present
functional
some
technical results that will be useful in establishing the properties of the
U {{Ct,Lt}
Al
Definition
X
Let
Suppose that
G
Z
and
be
X
Lemma Al
:
^
Z
be a vector-valued
mapping.
continuously (Frechet) differentiable in the neighborhood of xq with
is
Then xq
the derivative denoted by G' {xq).
maps
G X
Banach spaces and
is
said to be a regular point of
G
if
G' (xq)
onto Z.
Let
X
Z
and
be
Banach
spaces.
Consider the maximization problem of
P(u) ==max/(x)
(47)
subject to
90 (x)
< u
(48)
.
and
G (x) <
where f
X
:
—^
R
and
go
valued mapping and
go
is
convex,
Theorem Al
Let
is
—^
R
are real-valued functions
the zero of the
Banach space
that the solution at
Then
(4S).
Proof. This lemma
p.
X
and moreover
any multiplier of
Ozdaglar (2003,
is
:
(49)
pi
is
u
=
Z
.
and
G X
:
-^
Suppose that f
0, xq,
is
Z
is
is
a vector-
concave and
a regular point.
Let
fj,
be
a subgradient of P(0).
a direct generalization of Proposition 6.5.8 of Bertsekas, Nedic and
382) to an infinite dimensional maximization problem.
X
and
Z
be
Banach
spaces.
Consider the maximization problem of
P (u) — max / (x)
x&X
subject to
G {x) <
where f
:
X
vector-valued
^f
and
Then
that f
P (0)
is
a real-valued concave function
muppmg and
Suppose that xq
G
R
is
and
+u
is
and and
the zero of the vector space
G
X
Z and u
a solution to this program. Suppose also that
G
:
.tq
is
^'
is
Z
is
a convex
a perturbation.
a regular point of
are continuously (Frechet) differentiable in the neighborhood of xq.
is differentiable.
43
Lemma
Proof. From
subgredient and
is
Al,
follows that
it
there
if
has a unique
thus differentiable. Proposition 4.47 in Bonnans and Shapiro (2000) estab-
under a weaker constraint qualification condition than regularity,
lishes that
P
a unique multiplier,
is
this
problem has
a unique multiplier.
Theorem A2
X
on
is
Let
X
compact metric space, then the space of probability measures defined
he a
weak
a compact metric space with the
Proof. See Parthasarathy (1967,
45).
p.
topology.
a
Randomizations
We
To
next introduce randomizations to show concavity and differentiability of
original
maximization problem without randomization
and
(12) as stated in Proposition
A''
1
Ct
we suppress dependence on public
simplify notation, in this appendix,
-I-
:
^
K+ and
It
:
0' -^
{N {N +
1))*
and
By
[0,l\-
Y = F (Y, l) <
1))"'
+
2).
e Xf,
for A^^
let
of vectors in the set
(9)
9t
£ Q, where
9
histories
/i*.
The
subject to (10), (11),
is
at finite set (with
also a finite set. Consider next the functions
cx) is
definition, these functions assign values to a finite
<
t
can simply be thought of as vectors of
oo, thus
dG
ct {e')
< Y, Kt+i < Y and
(e*)
=
Xt
oo. Therefore,
Let X( be the set of
Xt
X/
(i)
{ci {6^)
all
(0*)
,
< Y,
Kt+i, x,}
(50)
a vector (of dimension
is
denote the zth component of this vector, and Tt be the dimension
(i.e.,
now construct
/
,
xt
such vectors that satisfy the inequalities in (50),
Tt
= {N [N +
l))"'
the usual Euchdean distance metric, dt {Xt,X')
Let us
maximize
to
({Ci, Ltj^g).
dimension. Moreover
/
[N {N +
<
t
of points in the set 0* for any
number
where
Recall also that
1.
elements). Therefore G* for any
Q*
is
W
+ 2).
=
(
Xj
X],=i
is
a
compact metric space space with
(-^'t ('')
~ ^t
(i))"
the product space of the Xj's
oo
(=1
I
Clearly the sequence {cf [6^)
to the subset of
Now
X. which
((9*)
therefore
topology
is
it
is
,
(e.g.,
(10), (11),
also
compact
oo
A't+i,^^}^^^ must belong to X. Lr fact,
satisfy (10), (11),
by Tychonoff's theorem
product topology. Since
X, and
,li
and
and
(12),
Dudlej', 2002,
must belong
denoted by X.
Theorem
2.2.8),
(12) are (weak) inequalities,
in the
it
X
product topology. Moreover,
X
is
X
is
compact
in the
a closed subset of
with the product
meterizable, with the metric
oo
d{X,X')=Y,^'di{XuXl)
£=1
44
(51)
some
(j)
topology
is
for
G
compact
X
is
X
This shows that
endowed with the product
X.
is
the set of probabihty measures defined over a compact metric space
weak topology.
in the
defined over
We
{-YJJ^g e X.
a metric space, and so
Prom Theorem A2,
is
X =
and
(0,1)
compact
in the
are concerned not with
information revealed up to
weak topology.
all
probability measures, but those that condition at
=
Let C
t.
{{cj.)
e Mr
<
:
<
c
<
c,
<
I
possible consumption-labor allocations for agents, so that P°° defined above
probability measures over C°°.
of
A''
+
Now,
Thus each element
consists of j¥
+
thus closed.
Consider
1
each
t
gN
and
C
(•
|
d*'^)
=
[((610
probability measures for each type
V =
UtenUe'eQ''^'
closed subset of a compact space
in the
for
6*'"-^
is
compact
(9,
C
^*-^),
|
for
V
let
is
t
on
be the set of
1}
the set of
is
all
be the space
[d^^^]
an individual with history
..., ((fi'./v
|
0*'^)] in a
given their past reports,
[^'~^]' w^'ich
(e.g.,
6*^\
e
1-tuples of probability measures on Borel subsets of
of reports 9^-\
7^°°
This establishes that the set of probability measures
a closed subset of
Dudley, 2002, Theorem 2.2.2),
P*
d''~^,
V^.
V
is
[0*-^]
and
is
Since a
compact
weak topology.
Finally, choosing
<p
< P
hi (51)
shows that the objective function
continuous in the weak
is
This establishes that including randomizations, we have a maximization problem
topology.
over probabilitj' measures in which the objective function
and the constraint
set
is
compact
in the
is
continuous in the weak topology,
weak topology, and thus there
exists a probability
measure that reaches the maximum.
Properties of W({Ct, Ltj^o)
We now
that in
established the
all
the proofs
main properties
we assume that
of W({Ct,L/})^Q).
The only
the solution to the maximization problem (9)
regular point. This needs to be imposed an assumption, since
the solution
since
if
is
it is
is
shifts
is
toa regular point
are not at regular points in this context are "non-generic," though
mathematical statement of
not a strong one,
is
increasing in Ls for a.ny
(i.e.,
solutions that
we do not present a
precise
property to economize on further notation and space).
this
U{{Ct, Li}'^^)
continuous and concave on
s
and
A°°_,
differentiable in {Ct,Li}
Proof. The above argument established that
and
at a
not at their regular point, a perturbation of the utility functions or the
production function should ensure that the solution
to (10), (11),
is
is
not possible to check that
indeed at a regular point. Nevertheless, this assumption
the solution
Lemma A2
additional restriction
in the
(12) over probability measures, a
45
nondecreasing in Cg and non-
oo
problem of maximizing
maximum
exists
(9)
subject
and U{{Ct, Lt}^Q)
is
therefore well defined.
To show
and (C\L^) and corresponding C°,C^-
concavity, consider (C°,L°)
We
have
[{uicJ-e)-u{cJ;e))C{dic,l)J)
= a
(u(c,
j
9)
/;
-
u(c,
/;
9))C{d{c,
I),
+
6)
-
(1
a)
I
(u(c,
6)
/;
-
u{c,
e)K\d{c,
I;
I),
9)
>
In a similar
is
feasible
and
way we can show that C"
it
gives the
same
satisfies (10), (11),
Next, note that the constraint set expands
U
must be weakly increasing
in
+
u{9)
utility as a(^^
if
(1
—
and
a)^^
C^ increases or
Cs and weakly decreasing in
Al
to conclude that
Lemma A3
A'^
is
U{{Ct,Lt}^Q)
is
(9)
decreases for any
is
defined over a
is
{Ct,Lt}Zo and
feasible for
{Cm}Zo
{a{Ct,Lt}'^Q
+
Now
respectively.
—
(1
for
A~
space.
we can use Theorem
and some C°,C^
any a £ (0,1) C"
a) {Cf^LJI^g), so that this set
is
=
a<°
+
feasible
(1
-
a)C^
non-empty. Moreover,
it
as well. Similarly, ("
the constraits on aggregate {Cf, Lt}'^^.
(Compactness) For any sequence {Cl',Ly}^Q G
exists a sequence {C"}f^o corresponding to
A°°, {Q",
I^lSo
~* {'^T'^'^JZo' there
{C^,L2]'^q, such that C" —>
incentive compatibility, aggregate constraints and feasibility, therefore
closed.
Banach
compact and convex.
since C°iC^ satisfy the incentive compatibility, constraints, C" satisfies
satisfies
therefore
{Ct,L/}^Q, completing the proof.
differentiable in
Proof. (Convexity) Consider {Ct,Lt}^o a^d {C',,L[}'^q e
for
s,
L.,.
at their regular point,
is
convex combination
u{9).
L,,
Finally, returning to the original topology, W({C/, L^JJ^q)
Given the assumption that the solution to
(12), this
Boundedness follows from boundedness
of
C
and
C°°i satisfying the
{C^,Lf°}'^Q e
A^
is
b
L.
Proof of Theorem 4
Proof. I'he
part
is
once we establish that
(5
first
By Theorem
2,
identical to
—
^p.
The
Theorem
3
and Parts
2
and
3 follow readily
rest of the proof est,ablishes that
the best sustainable
mechanism
/3
also solves in the
=
We now
3
in this case.
problem of maximizing
(15) subject to incentive compatibility constraints (32), as well as (16)
sequence {Ct,Lt}'^^^).
(/p
Theorem
and
(17) (for a given
write this quasi-Mirrlees program explicitly taking into ac-
count randomizations. To simplify exposition, we focus on randomizations across individuals
and ignore randomizations across
(e.g.,
different levels of Xt'S-
Applying Caratheodory's Theorem
Proposition 1.3.1 in Bertsekas, Nedic and Ozdaglar, 2003, pp. 37-38) to the problem of
46
providing a certain level of utility to each type at a given date,
finite
number
finite
consumption
each
771
of
consumption
G Mt
[0)
consumption
levels for
(6)
type 6 at time
by c™
level will
each type
levels for
be Mt
t
G
G
[9) for
G
(9
levels for
and denote the probability that
/,,
be given to an individual who has announced
program can be written
Let the set of these
Denote these consumption
Finally, let us denote the probability that an individual will
quasi-Mirrlees
on a
sufficient to focus
at each date.
[6).
and time
it is
his
this
type as 9 by g^
{9).
Then
the
be of type 9 hy n
(9).
as
oo
^i{Ct.Lt}^=
m{9)
m.{0)
max
qr'iO)u{cr'{0),l^
,.f'\9)
l^7r{9)l_^P^
eee
it,m(e)eM,(s),eee--"
{cr-<'',c<''},
—
(=0
'
,
m(fl)eM((e)
subject to the incentive compatibility constraints (equivalent of (32)), which take the form
oo
m{B)
m{e)
rnie)
E/5*^(^) qr'{0)uicr'{o)jr'ie)\
'qf'\9)u(^cf'\9),lf'\9)\9y
t=0
t=o
&
for all 9
Q
and 9 £ Q, and versions of
E E
E E
and
(16)
q-^'^
m(0)eMt(e)
(17),
which take the form
[9) cr^^) {9)
<
7r(0)(?r^^^6?)/r^'^0)
>
n
{9)
Ct
(52)
(53)
e€Gm(6l)6M((e)
for all
t.
Denote the multiplier of
~
implies that lAct ({Qi -^tlt^o)
straints be denoted
=
that r}{9,9*)
"^t-
(52)
by
^^^
^^^^
by vW^,^]. There
for all 9
<£
Q.
allocations in the quasi-Mirrlees
Then
Then, the
A,,
differentiability of
1
-|-
Yle^d* V (^*!^)
type 9*
will exist a
typically the highest type 0jv, such
,
program imply
=
^^^re is
u,
we denote
bounded by
feasibility
(9)
exists).
i^t
(9)
/J"
,
Fix a sequence {mt {9*)} such that mt {9*) G
a sequence clearly
(^)}t^o ^^^
Mt
\
9j
[9*] for all
in particular,
limsupt_ooUc (c^
case,
P*u*ri.
ip
t
(0, 1)
:
the proofof the theorem,
plinif^oo
Q'^^c
=
0}
=
and limsupf.
a
47
is
_gr^''^>0(such
IS
/J f=u
f=0
\
{9*)
is
\9*\
=
and hminft_oo"c
it*
sandwiched between
w*?7 or limsupj_^Qo/?~'At
Z^)
and u
rj.
a bounded stochastic sequence.
sufficiently large, \t
=
by
?*),/"""-^^'^(r
{9*) ,1^^
Therefore, either limsup(_^/3'~^\t
=inf{ g G
t
it
m,(0-)
Then, every subsequence of
u^ exist. This implies from (54) that for
and
is
>
1.
bounded, and
(54)
Xt.
a constant independent of time and
rn{9)
\
Lf}^g)
the first-order conditions with respect to consumption
Since the (stochastic) sequences {c( (0)}^o ^^^
continuously differentiable,
({Ct,
multipliers for the incentive competently to con-
0'uc{cf\9*),lf\9*)\9*)
Note that
W
estabhshing that
^ =
=
j3.
u*fj.
/3*u*Ti
In either
This completes
mt(6')
(^*),'
(cf
.
Appendix B: Proofs
for Section 9
Proof of Theorem 6
The proof
of this
Proposition
theorem follows the structure of the proofs of
The main
1.
difference here
is
Lemma
Theorem
1,
and
1
that instead of replacing the politician, citizens play
a null strategy of supplying zero labor.
Proof. Define the following combination. Denote
zero consumption to
xt^s y'-*'^)
=
^t-s
a\
=
and M/_5
f/^'"'')
Q, which
means that
and
7^
d,
if h'~'^
then r involves Xt
involves ^^
A
=
fJ
1,
difference
M,^s
==
{K[j_-^,c[
\
M*j
each citizen
for
/V"\ then
a\
=
and
for all
=
<fi
for all
G H', and
Lemma
1
Alt.
with type
6]
^
€
6q to
and
[0, 1]
>
for all s
We
0.
if
\
,
some other strategy that
/V"i
if
is
1",
x
^)
(0
=
to
show that there
time
tit
t
Suppose that
and has capital stock Kt, and
show that a deviation by an
involves supplying positive labor
individual,
output
F {Kt,el'),
since
6*
G
and there
will
^ 0,
>
1.
Then
(45) with A't+i
=
0,
for all s
(1
—
since
p) (since
be no labor supply for any
Now
/',
all
with x
{^'
imagine a deviation to a message that corresponds to
\
(^l
)
> X
[^
\
&'t
—
)
^
by definition. This
other agents are supplying zero labor.
Now
the seciueiitially rational stiategy of the government
since there will be
no production
will
generate
imagine the behavior
= q®
of the government at the last stage of the game, conditional on qJ^.^
and
i'
not profitable
is
obtains utihty u (0) /
i'
exists a
I
type in the continuation game).
positive labor supply, say
for all
it
cf
we need
a continuum of agents). Without the deviation,
from Assumption
/i'~\
/V'^ then
7^
(we think of an individual with positive measure s deviating, and take the limit £
there
=
if /i*~^
agents supply zero labor.
first
if
a= [a a®), for
h*'"^ = h}~^ then
the citizens,
e fP\ and
h*
the following strategy
we have that
/i'-
that
is
the government has announced a submechanism
foi" ^1^ *
(1) for
t,
=
cj
all
— ^^
Then
0.
=
Let h}
a®; (2) for the pohtician, F, such that
= Xt, Mt = Mt, and
= F {Kt, Lf) for all h'
from the proof with
alU:
for
^-^
i
>
for all s
sequentially rational continuation play in which
<^t+s
cf be the mapping that allocates
individuals irrespective of past and current reports.
all
combination would ensure vf
some
=
c.[
in future periods.
for all
is
to
G
i
[0, 1]
maximize
Consequently, the
utility-maximizing program of the government in the information set following the deviation
is:
max {l-a)v
subject to xt
+
(xt)
+
a
Jd't (c* {at (0*)))
history of reports up to time
of
[u {c[ (z* {a, {9'))))
^J
Assumption l\
t
dG*
[9^)
-^
{h (-' {o, {9')))
< F{Kt,sl'), where recaU
by an individual of type
this expression
is
concave in
48
c for
0*
\
9^)]
that
dG' (0^))
2* {ot {9^))
given strategy profile a.
any strategy
profile a, so the
,
is
the
In view
optimal
policy for the government in this information set
not consume
equally across agents,
itself)
This implies that as £ -^
/3
(u
-X
(0)
(0
strategy profile where
Now
all
0,
and thus the
< (u
/3)
cj (2'
i.e.,
(0)
-X
(0
I
consumption (what
to redistribute
=
(qj [6')))
for all
Ci
deviaition payoff of
-
C*'')) / (1
agents supply zero labor
is
i'
is
u
offers
Mf ^ Mt,
one imphcitly agreed
= a^
equilibrium, Qj+j
for all
G
i
[0, 1]
.x).
and
is
-
(
G Z^.
6'']
/'
\
First,
|
+
imagine the
beginning of time
at the
than the
t
Given the above-constructed continuation
>
for all s
a best response against this
is
— a®
Since maximal punishments are optimal, aj^^
deviation.
>
plan (Af,
in the social
(0)
(a, (S^'))
does
sequentially rational.
mechanism
a different
i-e.,
.;'
it
showing that a continuation
/3),
consider two different types of deA'iations by the government.
government
all s
-
^'')) / (1
I
^
0, c^
is
for all
i
£
and
[0, 1]
for
optimal against this deviation, implying that such a deviation would never be
profitable for the government.
Second, the government can deviate at the last stage of time
all
i
e
[0, 1]
and
>
for all s
1
the
is
maximal sequentially
rational
t.
Again aj+j
Thus the optimal deviation
further production.
punishment against such
lor the
government involves /\(+i
again exploiting the concavity of the government's continuation payoff in
is
ol^ for
Consequently, after any deviation by the government, there will not be any
a deviation.
constraint
—
equivalent
c,
=
0,
and
the sustainability
to:
(55)
00
:i
-
+ aE,+, [\
a)v[xt+,)
- x
\u (c, (3* (q, [Q'))))
{k {z' {ot (0')))
I
^t)]
dG'
s=0
>
max
(1
-
a) v
{xA
+
x[+c[<F{Kt,Lt)
r* and a*
we can
=
Theorem
(a*
uc, (9') dG' (6')
for
ah
t.
1
a') for
pair of strategy profiles
F and
a, exactly the
same argument
as
implies that there exists another pair of equilibrium strategy profiles
some
a'
such that F* induces direct submechanisms. Consequently,
write (55), in terms of a direct mechanism, which gives (46).
Finally, the
able
|
/
J
Now, given an equilibrium
in the proof of
a
same argument
mechanism
is
as in the proof of Proposition
1
implies that the best sustain-
a solution to maximizing (9) subject to (10), (11), and the sustainability
constraints of the government given by (46). a
Proof of Theorem 7
Proof. Suppose again that there are iV +
order of
skills,
and with respective
l
types,
i.e.,
9
probabilities {7ro,7ri,
49
=^
{^o, ^1,
...,
7r,v}.
•1 ^nJi ranked in ascending
Given the assumptions of the
[6')
theorem (and again suppressing /I'-dependence
for the best sustainable
mechanism
tlie
program
as:
N
oo
E^^' E
,.
.
U<,^n/^^^
we can write
to simplify notation),
-' [- (^.(^,0)
-X
(^^ (^,:)
I
^.)]
subject to the constraints
5^/3'
for all
E
=
i
for all
e,)]
X- (It {9:.)
I
> Y,0''
i (1
-X
ik (Or-i)
I
-
+ alf2^^[u
a) V i^t+s)
(ct+s {9r))
-X
ih+s
{9^)
\
9,)]
]\>V
(A'j,
and that
/:,
The
first set
Theorem
+
+>
A'i+i
5,
>
C( (0.;)
7r,-u (ct
{9i))
<F\Kt,}
\
for all
i
and
and
t
>
X(
TTJt
{9,.
I
(57)
for all
(58)
9,)
i=i
/
t.
of constraints, (56), ensure incentive compatibility for the citizens.
there
6,
Assumption
truthful revelation along the equilibrium path.
is
implies that
abled type, ^0, where type
we only need one constraint
i
for
mSiXx' +c' <F(Kt,Li) (1
for
~
'^)
''-'
each type other than the
could deviate to claim to be type
(^t)
+
<^Yl-i=o^i'^ic't {9i)),
and
i
—
The second
1.
finally,
(1998) and form ihe Lagrangian:
N
oo
^ =
Marimon
E/5'E^d"(ct(e.:))-Xa*(^,:)I^O]
t=0
i
=l
oo
'V
r
+ Y, /3Vt
(1
t=0
-
") vixt)
+ 5^
a
oo
i=o
Ku^n,!,
/x,_i)V
\
Af
{a
{9,))
-x
{U {9,)
\
9i)]
J
N
/
-
[u
i=l
[
-J2(i'i,^t
^
TT,:
\
{9i
\
0,)
/
7:=i
oo
C
,;=1
i
(=0
oo
C
t=o
1,
N
/
.¥
(=1
\
7:=i
50
dis-
set of
V{Kt,Lt)
=
the last set of constraints,
each date, impose the aggregate resource constraint.
again follow Marcet and
Given
This, together with
constraints, (57), one for each date, impose sustainability, with the definition
We
Lt)
and
t,
7:=o
one
(56)
9,)]
I
xt
for all
[« (c/ (^,-i))
l,...,N,
P*^'
s=0
-
[u (ct (0,))
\
^
.
where
and
i
For
the
is
the multiplier on the incentive-compatibility constraint of type
on the resource constraint at time
plier
all
A^
/3*
t
and
>
xt
>
for all
0,
terms and defining Aq
=
and
Xt
+
(1 4- ajd^) TTiu' {ct (d,))
+
(1
+
a/i,)
-
[fif-
-
(A,;
we can take
-
+
{X,x' (k
[0,,)
e,)
I
extended real
First
line.
-
an allocation with
—
>
r/*,
above maximization problem with
Combine
we have that
1
,,
Both
that
|J^
—
>
00 implies that as
+
/li^
—
>
oo,
and
i
+
—
>
c*
»
we must have
It
this also implies that as
t
—
>
oo,
oo,
(59)
t,
0.+i))
\
0, ...TV
for all
and ah
(60)
t,
(61)
t,
(62)
possesses a unique limit point on the
then (59)- (62) establishes that there
for all
i,
and
—
xi
—
oo.
>
x*
>
,
which distortions
in
{9^)
>
_
~
S
Mt
fact that
and
Mt+i
>
x;
_
-
<
fj-t+i-
i'
^
i
=
—
/i(_,_i
—
>
/ij
"
/
>
/ij
—
and that A^v+i
oo.
»
=
1
+
V
»'
,(.n\
^^'^'
•
v'{xH-i)
>
fact that /ij^j
The hypothesis
|J,^> 0.
to obtain:
=
>
u' (c, {9,))
oo,
|c(
(&,)
0, ,..TV.
From
for
=
disabled).
Cj {Oi) |
any solution to the
0,
--
by the
/i-j
t
exist
To obtain a contradiction suppose that
+ i^lLZ^rt'
TT.,'
i
and ah
t.
show that there does not
(Ct {9,))
{9i) [ l*
become
9q would claim to have
<
jj*
»
it
ai^i-t)
for all
c*
J,
[0,)
=
i
to
oo implies that as
establishes that ct{9.;)
implies that
—^
ih^L^u'
TTiil
fact that
t
some
(59) for
n' {Ct {Oi))
Oi 7^
—
/n^
sides of this equation are strictly positive
Next combine
The
for all Cj
^70^7)
.
for all
for all
rjt
and using the
^)
^N J
+
—
X,+ix' ih
0, ...TV
.
/i(
(59) for type TV with (62)
(by definition),
^
a) v' [xt)
Ct {Oi)
we need
the proof,
-
=
i
+ 1-j^Fk [Ku Lt) - r^m^x =
disappear as claimed in the theorem.
To complete
for all
9^)
\
{Ki,Lt)
iltT^iFi
suppose that
??,
=
7?t7r,:
a nondecreasing sequence, thus
is
exists
-
Lt)
(1
p.t
Recall that {fifj^Q
for
0, yield:
{Ci [6,))
Vk {Ku
Mt-i)
=
- Xi+i)u
n,x' ik {Or)
-
>
c,, (6*,:)
first-order conditions, which, after canceling out
and A/v+i
^it-i) tTjVl {Ki, Lt)
(Mt
ha,ve left the constraints that
the multi-
implicit.
t
>
(Oi)
Ct
and we
t,
z, /3''r]^ is
all i
—
(1-1-
ci {0,/)]
51
i
=
{9,))
(64)
-^
0.
This argument then
the freedom of labor supply, this also
1, ...TV,
since otherwise at
From Assumption
for all
(r,
a^,J
0, ...TV
and
2
,C(
some point
all
and the resource constraint,
J.
0.
NoM', suppose
all
i
and
all
>
t
first
t'
We
mechanism.
for
that
-
l*
some
t'
have that
and
<
We
oo.
for all
>
t
continuation utility of the government
However, by hypothesis,
Moreover, from
would prefer
Ct'-i
(64), Ct'^i
{9,,)
to deviate at
equal consumption across
{9-,)
t'
all
=
c*
show that
will
t'
,
is (1
>
are reached in iinite time,
Ct[9i)
-
u
a)
=
(0) / (1
for at least
some
i
and
remain at
(1
—
a)
u
(0) / (1
—
individuals
Cf-i
(i.e.,
and
and
so there
z,
{9i)
5).
—
—
Ct {9i) [
and
but
[9;) [ 0,
/(
(64) implies that, as long as au' (0) 7^ (1
—
=
q
[9,)
a) v' (0),
>
and some
i
—
(0
so the
T) =
0)-
-
t'
1.
^
This
C(,_j).
Ct'-i {6i') for
onwards would
0, it
still
proves that there
Hence,
in finite time.
it
Then, combining (63)
t.
j 0,
/^^
t'
>
t/
for all
^-,.|.i
0,
the output between Xt and
=
c*
=
Xt
and x
(since Cf-i {9;)
1
and
for
implies that the government
Since this argument applies for any
/*
—
{9i)
positive output at
is
c*,_j for all
/,'
=
w (0)
5) (since
and
i
continuation utility from
its
cannot be a sustainable mechanism that reaches
must be the case that
for all
(0.,;')
in the original allocation),
i'
=
for some i and i'. This
^ Cf'_i
— 1 to £,ti^i = 1, and redistribute
deviation will necessarily increase government utility at
all
-
Cfi
cannot be part of a sustainable
this
[Oj)
U
i.e.,
contradicting
—
>
/j.^
00,
and thus establishing the theorem.
Proof of Theorem
Proof. In
8
this case, the best sustainable
mechanism can be represented
as the solution to the
following maximization problem:
MAX2
:
max
U^^'^=
{c,,(-),(/.(-)-n,A',,H-i}^o
subject to the
E
initial capital stocl-c A'o
X^<5^ {(1
>
E X] /?'«(c.
0, (10), (11)
- a)v{xt+,) + au
/tM
(('?'')),
I
^j)
/=o
(c, {{0'''))
and the modified sustainability constraint,
Jt
{9''')
\
>v{F{Kt,Lt)),
9"^}
(65)
,s=0
which again takes into account that
after deviation there will
be zero labor supply and thus
the highest continuation value of the government after deviation
is
denote the Lagrange multipliers on the sustainability contraint by
and rewrite
MAX2
MAXo
U'^-'*^^
:
^''
[F {Kt,Lt)). As
t^V'ti
with
^.^
=
before,
/Hj_i
+
il^i,
recursively as
max
E X:^'n
(c, {{9^'^')).k
{9'')
Lt=o
+ J2
v
{l^t (1
-
a) vixt)
-
(M,
- ,.H-MF{Kt.,
f=0
r-,2
Lt))}
i
9])
+
f,,5'au (c, {{9"^))
,1,
(^'*)
subject to (10) and (11).
by
{a**}-
which
is
Then
Denote the Lagrange
inultipliers in the solution to this
as in the text, consider the quasi-Mirrlees
program corresponding
to
problem
MAX2,
given by
Y,{P' +
{ci.{0'),h{n}T=,
^H5'a)u{ctie^-*),lt{0''')\Ol]
t=Q
subject to (11), and the two additional constraints,
j ct{e')dG{e')<Ct,
and
Jlt{9')dG{9') >Lt.
Here
U {{Ct, Lt}'^^;
except that
it is
arguments as
{Mt }(^o) ^^^ exactly the
as
U
{{Cf, Li
jgg)
in the text,
sequence of Lagrange multipliers. Then, with the same
also a function of the
in the text, the
same interpretation
maximization problem
MAX2
becomes
00
max
U{{Cu L,}^o;
{m* ll^o)
+
E
^'
{a'* (1
-
a) v{x,)
-
{,,;
- f,UHF{I<u
Lt))}
subject to (10) and (65).
Proceeding as
in the
Using the definition of
(1
which
is
-
proof of Theorem
if,
this
3,
we obtain
can be rewritten as
=
a) 6'v'{x*)
Mt
'^*
<
-
M<+:
'^'+'
identical to (31) in the proof of
=
7;
(1
-
Theorem
3,
yields the conclusions in the theorem. In particular,
and distortions disappear.
If,
on the other hand, f >
and asymptotic distortions remain,
a
53
,,
, rt+i
,,
a) 5'+'v'{x*)
and the
5,
i
^ 00,
rest of that proof applies
'-p
=
5,
/j.(_^j
>
fi*
when
as
then we must have
and those
t/'j
>
—
i —
jU*
as
and
>
jl*
>
00
A
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