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OBWSY
HB31
.M415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
MARKETS VERSUS GOVERNMENTS: POLITICAL
ECONOMY OF MECHANISMS
Daron Acemoglu,
Michael Golosov, Aleh Tsyvlnski
Working Paper 06-09
April 24, 2006
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research Networl< Paper Collection at
http://ssrn.com/abstract=898685
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUN
2
2006
LIBRARIES
Markets Versus Governments: Political Economy of
Mechanisms*
Daron Acemoglu
Michael Golosov
MIT
MIT
First Version:
Aleh Tsyvinski
Harvard
September 2005.
Current Version: April 2006.
Abstract
We
study the optimal Mirrlees taxation problem
ductivity or preference) shocks.
the mechanism
centralized
power,
is
operated by a benevolent planner with
mechanism can only be operated by a
who can
our analysis
is
in a
dynamic economy with idiosyncratic
(pro-
In contrast to the standard approach, which implicitly assumes that
therefore misuse the resources
full
commitment power, we assume that any
self-interested ruler/government without
and the information
it
An
collects.
that there will be truthful revelation along the equilibrium path (for
commitment
important result of
all
positive discount
which shows that truth- telling mechanisms can be used despite the commitment problems
factors),
and the
different interests of the
government.
patient as the agents, the best sustainable
gregate distortions arising from political
Using this
tool,
mechanism leads
we show that
if
the government
to an asymptotic allocation
economy disappear. In
contrast,
when
is
as
where the ag-
the government
is less
patient than the citizens, there are positive aggregate distortions and positive aggregate capital taxes
even asymptoticallj'. Under some additional assumptions on preferences, these results generalize to the
case
when
the government
is
benevolent but unable to commit to future tax policies.
We
conclude by
providing a brief comparison of centralized mechanisms operated by self-interested rulers to anonymous
markets.
Keywords: dynamic
incentive problems,
mechanism
design, optimal taxation, political economy,
revelation principle.
JEL
Classification: Hll, H21, E61, P16.
*We thank Manuel Amador, Marios
Angeletos, Abhijit Bancrjee,
Tamer Basar, Timothy
Besley, Olivier
Blanchard, Ricardo Caballero, V. V. Chari, Mathias Dewatripont, Caroline Hoxby, Larry Jones, Patrick Ke-
Guido Lorenzoni, Asuman Ozdaglar, Christopher
Muhamet Yildiz and seminar
Canadian Institute of Advanced Research, Carnegie Mellon,
hoe, Narayana Koclierlakota, Jonathan Levin, David Levine,
Phelan,
Andrew
Postlewaite, Vasiliki Skreta, Robert Townsend, Pierre Yared,
participants at Boston College, Boston University,
Cornell-Penn State Macro Conference, Federal Reserve Bank of Minneapolis, Federal Reserve Bank of PhiladelMacroeconomics Conference, Federal Reserve Bank of Richmond, Georgetown, Harvard, MIT, Princeton,
pliia
Rochester, Stanford,
useful
comments and
UCLA,
Yale, University of Iowa, University of Kansas,
to Gleg Itskhoki for excellent research cissistance.
and University of Pennsylvania
for
1
Introduction
The
first-generation approach to public finance, best exemphfied
ation, sought to determine the optimal policy of a benevolent
given set of
fiscal
or regulatory instruments.
made major
Mirrlees, has
by models of Ramsey tax-
government
in
a world with a
The second-generation approach, pioneered by
progress over this approach by explicitly modeling the choice of
tax instruments that the government can use.^ This hterature has modeled the informational
problems restricting the potential tax-transfer programs, and formulated the determination of
optimal tax and transfer programs as one of mechanism design.^
Some
of the theoretical limitations of the second-generation approach have long been ap-
parent, however. First, in intertemporal settings, this approach assumes that the
mechanism
designer (government or planner) can commit to a dynamic mechanism, even though such
commitment
is
ex post costly. Second,
it is
assumed that there
is
a body (a "benevolent gov-
ernment") that can operate the optimal tax-transfer program (mechanism), even though the
lessons of the political
maximize
economy
literature are that
welfare, but have their
own
governments or politicians do not simply
selfish objectives,
such as reelection or personal en-
richment.^ This paper aims to contribute to a potential third-generation approach to public
finance where both the informational constraints on tax instruments and the incentive prob-
lems associated with governments, politicians and bureaucrats are taken into account.
this purpose,
self-interested
Two
tainable
we
investigate
how mechanisms work and should be designed
For
in the presence of
and time-inconsistent governments.^
questions motivate this analysis.
mechanisms
—
i.e.,
First,
we would
like to
understand whether sus-
optimal tax- transfer programs in the presence of these additional
'See Mirrlees (1971) for the seminal reference and Baron and Myerson (1982), Dasgupta, Hammond and
Maskin (1979), Green and Laffont (1977), Harris and Townsend (1981), Myerson (1979), and Holmstrom and
Myerson (1983) for some of the important papers in the early literature. Albanesi and Sleet (2005), Battaglini
and Coate (2005), Golosov and Tsyvinski (2004), Golosov, Kocherlakota and Tsyvinski (2003), Kocherlakota
(2005) and Werning (2002) consider applications of the Mirrlees framework to dynamic taxation.
^Given this formulation of the optimal tax-transfer program as a mechanism, we will use the terms "optimal
tax-transfer program" and "mechanism" interchangeably.
^For general discussions of the implications of self-interested behavior of governments, petitions and bureaucrats, see, among others, Buchanan and TuUock (1962), North and Thomas (1973), North (1981), Olson (1982),
North and Weingast (1989), and Dixit (2004). Austen-Smith and Banks (1999), Persson and Tabellini (2000)
and Acemoglu (2005a) provide introductions to various aspects of the recent developments and the basic theory.
Another potential difficulty with centralized system is that they may involve excessive communication relative
to trading systems. See Segal (2005) for a recent model developing this insight.
One can think of 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 this may be a useful
modeling tool, it does not circumvent the substantive issues raised here: the party entrusted with taxes and
transfers has neither the same interests as those of the citizens nor much commitment power. Naturally, the
best mechanism we characterize can be represented as a solution to this fictional mechanism design problem.
constraints
full
—look similar to
commitment.
assume a benevolent planner and
"Mirrlees" mechanisms, which
they do, then we can have more confidence in the mechanism design ap-
If
proach as a tool to analyze the practice of policy design as well as a normative benchmark.^
Second, in the presence of self-interested and time-inconsistent government behavior, anony-
mous market
allocations cannot always be replicated
by centralized (sustainable) mechanisms.
This opens the door to a theoretical analysis of the relative costs and benefits of centralized resource allocation
intervention,
To
methods versus those that preserve anonymity and
and we would
like to
take a
highlight the problems that arise
planner with
full
commitment,
it
is
limit
government
step in this direction.
first
when we depart from
the
benchmark
useful to start with Roberts' (1984)
of a benevolent
example economy,
where, similar to Mirrlees (1971), risk-averse individuals are subject to unobserved shocks
But
fecting the marginal disutility of labor supply.
model, the economy
Under
full
is
repeated
T
from the benchmark Mirrlees
differently
times, with individuals having perfectly persistent types.
commitment, a benevolent planner would choose the same allocation
which coincides with the optimal solution of the
ernment without
full
commitment cannot
static model.
showed that
from exploiting the information that
refrain
game between
as discounting disappears
and
game
involves the highly inefficient
at
dates, supply the lowest level of labor
all
example shows the potential
case of
full
T —+
outcome
in
oo, the
which
can
has
Roberts
unique sequential equilibrium of this
types declare to be the worst type
all
arise
it
This turns the optimal
the government and the citizens.
and receive the lowest
inefficiencies that
at every date,
However, a benevolent gov-
collected at previous dates to achieve better risk sharing ex post.
taxation problem into a dynamic
af-
level of
consumption. This
once we depart from the unrealistic
commitment, even with benevolent governments.
Our benchmark economy
incorporates the lack of
paper, but also assumes that the government
is
commitment present
self-interested
in Roberts' (1984)
and maximizes
its
own
utility.
This latter assumption brings our model closer to the political economy hterature, where issues
of conflict of interest
purposes because
it
and
credibility of policy are central. It
simplifies the structure of the
is
also particularly useful for our
dynamic game
relative to the case with a
benevolent time-inconsistent planner. At the end of the paper, we generalize some of our results
to a situation where the government has an arbitrary degree of benevolence (thus nesting the
fully-benevolent time-inconsistent government).
Our main departure from Roberts'
(1984) framework
self-interested nature of the government.
that instead of a finite-horizon
we look for the allocation that maximizes the ex ante
economy and commitment constraints introduced by the
^In line with this objective, throughout the paper
utility of the citizens (agents) subject to the political
is
economy, we study an infinite-horizon economy, where individuals can use punishment strategies against the
government. This enables us to construct a sustainable mechanism, defined
as an equilibrium tax-transfer
government
for the
sustainable
if
(i.e., it
mechanism
program that
satisfies
is
both incentive compatible
for the citizens
and
a sustainability constraint for the government). The (best)
gives a fraction of the output to the
government in every period, and
the government deviates from this implicit agreement, citizens switch to supplying zero
bor, implicitly punishing the government.^
first
important
The
infinite-horizon setup enables us to prove our
a version of the revelation principle, the truthful revelation along
result, that
the equilibrium path, applies, irrespective of the discount factors of various parties.
that truthful revelation principle applies only along the equilibrium path
it is
la-
actions ofT the equilibrium path that place restrictions on
what types
is
The
fact
important, since
mechanisms are
of
allowed (these are encapsulated in the sustainability constraints).
The
truthful revelation along the equilibrium path enables us to write the problem of
finding the best sustainable
mechanism
characterize the solution to this
sum
the
that the
is
maximized subject to the standard incentive compati-
and two additional resource constraints
of total labor supply in the
sum
of total
economy be no
less
also optimizes over the sequences of Ct
identical to the full-commitment
is
is
problem
(i.e.,
Lt and Ct).
of characterizing the difTerences
commitment Mirrlees mechanism
in
Ct-
When
the mechanism
dynamic Mirrlees problem.
We
a solution to a quasi-Mirrlees problem, and distor-
tions resulting from the self-interested behavior of the
way
than some amount Lt and the second
and Lt subject to the aggregate resource constraints,
show that the best sustainable mechanism
of this quasi-Mirrlees
at every date; the first requires that
consumption be no greater than some amount
the quasi-Mirrlees problem
We
program by defining a quasi-Mirrlees problem, where the ex
ante expected utility of the citizens
bility constraints
as an infinite-dimensional maximization problem.
government only
affect the
parameters
This formulation therefore gives us a clean
between the best sustainable mechanism and the
full-
terms of the aggregate distortions caused by the former
relative to the full-commitment Mirrlees allocation.
The
other main results in the paper concern the characterization of these aggregate dis-
tortions. First,
we show
that at the initial date, there will always be further distortions in the
punishment strategies that sustain this equilibrium are not "renegotiation proof
This is
with many other analyses of repeated games, so we do not view this as a special shortcoming of
our approach. Moreover, in practice, citizens have many other recourses against governments that misbehave,
Clearly, the
.
common
them out
of office, and these actions will have the same impact as supplying zero
do not incorporate these possibilities to simplify' the analysis in this paper.
In Acemoglu, Golosov and Tsyvinski (2006), we study the equilibrium of an economy without any informational
restrictions on tajces and transfers, where the citizens have the option of replacing the current government. This
enables us to highlight the distortions arising from "pure political economy".
including voting or throwing
labor in the punishment phase.
We
sustainable
initial date,
mechanism
relative to the full-commitment Mirrlees
the sustainability constraint
is
mechanism.
Intuitively, at the
never slack and implies that any increase in output
has to be associated with increased government consumption. This increases the opportunity
cost of increasing output,
and leads to a reduction
in labor supply
and capital accumulation.
Second, we provide tight conditions under which these further distortions disappear or persist
over time.
we prove that when the government
In particular,
is
as patient as, or
more
patient than, the citizens, the sustainability constraint of the government eventually becomes
In the absence of a binding sustainability constraint, aggregate distortions disappear,
slack.
and the marginal products of labor and capital asymptotically converge to the marginal rates
of substitution for the agents; consequently, the limiting allocation can be implemented with a
structure of taxes for labor similar to that in the full-commitment Mirrlees
aggregate capital taxes. ^
When
the government
is less
economy and zero
patient than the citizens, the results
are very different, however; aggregate distortions never disappear,
there are positive aggregate capital taxes. This last set of results
is
and even
in the long run,
important, since
it
provides
an exception to most existing models, which predict that long-run taxes on capital should be
equal to zero
The
tions
(cfr.
footnote
7).
results for the case of a self-interested of
on the
utility functions of citizens.
utility functions that are separable
any
utility function of the
government are derived under general assump-
We also show that when individuals have instantaneous
between consumption and
leisure, similar results
apply for
government, in particular, in the case where the government
is
fully-benevolent but time-inconsistent.
Finally, since with a sustainable
mechanism part
of the output hcis to be given to the
government, the anonymous market allocation, without the government, cannot always be
achieved by a centralized mechanism. This raises the question of
are preferable to
issue;
anonymous markets.
We
centralized
mechanisms
conclude the paper with a brief discussion of this
we show that anonymous markets with limited insurance and redistribution may be ex
ante preferable to centralized mechanisms, and that this
institutional controls
This paper
original
is
is
more
likely
when
there are worse
on government behavior and a lower discount factor of the government.
related to a
number
of different strands of research. These include both the
and the more recent applications of the mechanism design approach to the optimal
taxation problem already mentioned in footnote
^This result
models,
when
e.g.,
is
1.
The major
difference between our
therefore similar to that of zero limiting taxes on capital in the first-generation
Chamley
(1986) or
Judd
(1985), but
is
work and
Ramsey-type
derived here without any exogenous restriction on tax
instruments (see Kocherlakota, 2005, for the zero capital tcix 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.
these papers
paper
3.
is
that they assume a benevolent government and
related to the recently burgeoning political
is
What
economy
distinguishes our paper from this literature
is
full
commitment. Secondly, our
literature
mentioned
in footnote
the explicit modeling of the incentive
problems on the side of the individuals as well as our focus on best sustainable mechanisms.^
Our
analysis,
is
also related to
work on optimal taxation with time- inconsistency,
for
example,
Chari and Kehoe (1990, 1993), Phelan and Stacchetti (2001), and Sleet and Yeltekin (2004).
As
well as Roberts (1984),
most
closely related to our research
is
the recent important
paper by Bisin and Rampini (2005), who also consider the problem of mechanism design
without commitment in a two-period setting.^ Bisin and Rampini extend Roberts's analysis
and show how the presence of anonymous markets acts
government, ameliorating the commitment problem.
the lack of
commitment by the
self-interested
distinction between the two approaches
is
as
an additional constraint on the
This lack of commitment
government
that our model
is
in
is
related to
our model. The most important
This enables us to
infinite horizon.
construct sustainable mechanisnis with the revelation principle holding along the equilibrium
path, to analyze substantially more general environments, and to characterize the limiting
behavior of distortions and taxes.
The
rest of the
paper
is
organized as follows. Section 2 describes the basic environment.
Section 3 starts the analysis of sustainable mechanisms. In this section,
of constructing sustainable
mechanisms and prove a version
mechanism
quasi-Mirrlees program. Section 5 characterizes the best sustainable
illustrate the
mechanism without any
restrictions
main
ideas.
on the
set of
set
up the problem
of the revelation principle. Section
4 formulates the problem of characterizing the best sustainable
but restrictive case to
we
as a solution to a
mechanism using a simple
Section 6 characterizes the best sustainable
mechanisms and preferences. Section 7 ex-
tends our analysis to cover the case of fully-benevolent, but time-inconsistent governments.
Section 8 briefly compares allocations with
anonymous markets
to those under sustainable
mechanisms. Section 9 concludes, while the Appendices contain some technical material necessary for the anatysis as well the proofs not provided in the text.
Kremer and Mian (2003)
and markets, but do not derive the costs of governments from
the centralization of power and information in the process of operating a mechanism.
'See also the work by FVeixas, Guesnerie and Tirole (1985) on the ratchet effect and recent work on general
mechanisms without commitment, for example, Bester and Strausz (2001), Skreta (2004), and Miller (2005).
*In this context, Hart, SUeifer and Vishny (1997), Chari (2000) and Acemoglu,
also contrast the incentive costs of governments
Demographics, Preferences and Technology
2
The model economy
is
infinite horizon in discrete time.
measure normalized to
citizens with
1
and a
ruler. '^
populated by a continuum of
It is
The ruler/government can be thought
of as a single agent or as a group of agents such as a bureaucracy, whose preferences can be
consistently represented
=
Let
6i
{Oq, 9i,
...,
by a standard von Neuman-Morgenstern
07V
be a
}
corresponds to "higher
0^
finite
skills"
ordered set of potential types, with the convention that
than
0,:_i,
and
in particular, Oq
be the T-fold product of 0, representing the
We
with each element belonging to 0.
drawn from 0°° according
0°°.
The
i-th
element of
standard notation
time
0''*
to
9], is
Let
the worst type.-^^
is
set of sequences of length
T =
l,2,...,oo,
think of each agent's lifetime type sequence 0°° as
some measure
0*'°°,
utility function.
the
/Li°°.
Let
9^'°°
be the draw of individual
of this individual at time
skill level
to denote the history of this individual's skill levels
up
to
t.
We
i
from
use the
and including
and make the standard measurability assumption that the individual only knows
t,
at time
t.
Since this will be a private information economy, no other agent in the
will directly
observe this
properties of individual
history.-^"
skills.
process.
We
Both
iid
draws from
assume that each individual's
according to the same measure
so that there
is
/x°°
economy
This structure imposes no restriction on the time-series
in every
temporal dependence are allowed. For concreteness, one
Markov
0''*
period as well as arbitrary
may wish
to think that 9] follows a
lifetime type sequence
and independently from the draws
no aggregate uncertainty
in the skills distribution.
of
is
all
drawn from 0°°
other individuals,
In addition, to simplify
the notation, we also assume (without loss of generality) that within each period, there
is
an
aggregate invariant distribution of types denoted by G.
The instantaneous
utility function of individual
i
at time
t is
given by
u{clll\9l)
where
cj
>
is
(1)
the consumption of this individual and
1}
>
is
her labor supply.
This
^"The continuum assumption implies that whenever we think of deviations, these should be by an individual
e, and then we should take -the limit as e — 0. Moreover, equilibrium statements should
be read as "almost-everywhere" These technical details do not matter except in the proof of Theorem 5.
^^Finiteness of
is adopted for simplicity and without loss of any economic insight. The more general case
where
is a compact interval of R+ introduces a number of additional technical details, not central for our
with positive measure
>
.
analysis.
^^This means that there exists a set of nested information sets (sub-sigma
fields)
that the individual only knows the information contained in
representing each individual's
at time t. In particular, let
be a probability space and {j^ t E Z+} be a filtration, i.e., a collection of sub-sigma
fields of :F, such that J^ C Tl for all t' > t. Let 0' be the set 0°° truncated at t. Then S''* € 0' and all
decisions taken at time t by individual i must be J^ -measurable.
information
sets, so
the triple (0^°,
J-", /li°°)
t
J-^
formulation
We
general enough to nest both preference shocks and productivity shocks.-'^
is
assume that labor supply
of
an individual with
skill
9
comes from a compact
set, i.e.,
e
l\
Wi.0)].
Assumption
function) For
(utility
1
all
6*
G 9, u(c,
continuously differentiable and jointly concave in c and
increasing in
(c,
I
Q
9 E
\9) /
\ui (c,
^
and 9
probability after
The
first
l\0)\
9q.
any
is
type
increasing in 6 for
and
Moreover,
and
:
is
E+
x
— M w
[0,r(6')]
twice
*
non- decreasing in c and non-
We
full support)
/x°°
all c
and
I
and
have T{9q)
9
all
—
&
E
and
has full support in the sense that
ui.
9\
I (9)
—
—
I
<
oo for
9q has positive
history.
two assumptions are standard. Assumption 3 states that
supplying positive labor
"disabled"
0)
(single crossing) Let the partial derivatives of u be denoted by Uc and
Assumption 3 (worst
all
|
I.
Assumption 2
Then Uc
I,
/
impossible. This suggests that
is
—unable to supply any labor
for the
we can think
worst type,
^o,
of the worst type as
at that date. It also requires pP° to
have
full
support in
the sense that any individual can become disabled at any point. This assumption will simplify
the analysis of sustainable mechanisms by making
actions where
all
it
possible to have off-the-equilibrium path
As described
types supply zero labor.
in
Remark
1,
this
assumption can
be replaced by an alternative one, described below, which imposes "freedom of labor supply"
directly.
The advantage
of
Assumption
3
is
that
it
leads to the freedom of labor supply as an
equilibrium outcome.
Each individual maximizes the discounted sum
(0, 1),
so their objective function at time
E E/5^-
E Y,P'^{<^t+sA+s\o + S)
E
\;\J^t\ o'^
the history
0*'*
G
t
9''\h*
s=0
s=0
where
j3
t \s
n
H+s)
of their utility with discount factor
^
['l^'"')
denote the expectations operator conditional on having observed
^*]
in addition to
any public information up to time
i,
captured by
ft*
(discussed
further below).
The production
side of the
economy
is
described by the aggregate production function
Y^F[K, L)
where
K
is
capital
and L
is
labor.
We
(2)
assume:
^In particular, productivity shocks would correspond to the case where
u
(cj,,ij
|
9J)
=u
{c\,l\/9X).
Assumption 4 (production structure) F
tiable in both of its
and
to scale
arguments, with derivatives denoted by
satisfies the
Inada condition,
capital fully depreciates after use,
Both the
F {K, 0) =
and
of labor in the
of output that can be
maximum amount
The Inada
economy
produced
Y
6
is
(0,
assume
Assumption
>
v' {x)
for
:
M_|_
.
Moreover,
is
essential for pro-
condition, together with the fact that
bounded, implies that there
oo) given
hy
t is
Y = F
is
a
maximum
{Y,L), where
L
is
the
given by
M.^
^
IR is
the ruler's instantaneous utility
refers to the expectations operator conditional
on public information at time
that:
5 (ruler utility) v
x €
all
M+ and
v (0)
is
=
twice continuously differentiable, concave,
0.
Moreover 5 e
Notice also that the ruler's discount factor,
citizens,
L e
=
where x denotes government consumption, v
We
all
oo
.3
t.
for
of total labor.
'
and E(
=
exhibits constant returns
0.
In addition, the ruler's (government's) utility at time
function,
and Fi,
depreciation assumption and the assumption that labor
full
maximum amount
amount
Fi<:
{K, L)
lim/<-_,oo Fj<c
duction are adopted to simplify the notation.
the
increasing and continuously differen-
is strictly
5,
and
satisfies
(0, 1).
is
potentially different from that of the
/3.
Sustainable Mechanisms
3
Since the government both lacks
for its
(e.g.,
own consumption,
commitment power and has the
the interaction between the citizens and the government
Roberts, 1984, Chari and Kehoe, 1990, 1993).
the equilibrium of this
best sustainable
ability to expropriate
Our purpose throughout
game between the government and the
citizens,
is
is
output
a game
to characterize
corresponding to the
mechanism, meaning the sustainable mechanism that maximizes the ex ante
utility of citizens.-^'*
^''
Since
selection
we are dealing with a dynamic game, our focus on the best sustainable mechanism is essentially a
among the many equilibria. Alternatively, one can think of the "social plan" as being designed by the
citizens to
ment
maximize
their utility subject to the constraints placed
(see, in particular,
by the
self-interested behavior of the govern-
the last paragraph of the Concluding Remarks, and also Acemoglu, 2005). In addition,
throughout the paper we focus on perfect Bayesian equilibria (see Definition 1) and do not impose renegotiationis both technically and conceptually difficult
and also would put more constraints on the "best sustainable mechanism").
proofness (focusing on the renegotiation-proof subset of equilibria
This section sets up the details of the game between the government and the
proves Proposition
1,
which provides an infinite-dimensional maximization problem
characterization of the best sustainable mechanism. This proposition
and
citizens,
for the
then used to determine
is
the form of the best sustainable mechanism and the resulting distortions. This section will also
establish a general result
Theorem
5 below),
for infinite-horizon
on truthful revelation along the equilibrium path (Theorem
which
is
used here to prove Proposition
1,
but
is
also
1,
of independent interest
games and mechanism design problems without commitment, and shows
that these problems can be analyzed without giving up the revelation principle.
3.1
The Game Form Between Government and
There
is
and the
a number of alternative ways of specifying the
with identical results. Our choice here
citizens,
Citizens
game form between the government
is
motivated to maximize similarity
with Roberts' (1984) model.
We
define a
submechanism
(or
i-mechanism) as a subcomponent of the overall mechanism
A
between the government and the individuals.
given date. In particular,
let
submechanism
what happens
specifies
Zt be a general message space for time
t,
at a
with a generic element
zt}^ This message space may include messages about cvurent type of the individual, 9^ G 9,
'•i t — l
and past types 9
€ Q^"^ (even though the individual may have made some different reports
'
about
his or her tj^pes in the past),
and might
also include other messages.
t
Let Z*
=
Yl Zs and z^ denote a generic element of Z^. In addition, let ht G
Ht be some
t
publicly observable aggregate variable, and
variable up to time
randomization, and
Here
t.
is
ht
/i*
G
iT'
=
H -^s
s=0
denotes the history of this
could be payoff relevant or simply a "sunspot" variable used for
not affected by the actions of the agents. These histories are introduced
to allow for randomizations.
A
submechanism
consists of
two mappings,
i.e.,
Mt =
(ct,
k
)
such that
ct
:
Z^ x H^ -^
E+
assigns consumption levels for each complete history of messages and public histories, and
It
:
Z^ X H^
^
[O, l\
assigns corresponding labor supply levels. -^^ Given
Assumption
3,
any
't
'^More formally, 5', 6 and zt have to be J-j -measurable as defined in footnote 12.
^^The mechanisms we describe here allow for general message spaces, but impose two restrictions. First, they
are non-stochastic. This is only to simplify notation in the text, and everywhere we can replace the mappings
in the text with Ct
Z' x ff ' -* A (R+) and [j Z' x if'
A ([O, []), where A (J) denotes the set of probability
measures over J. In 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 individual's 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
:
:
^
submechanism must allow
submechanisms that
some messages which
for
satisfy this restriction
whenever such conditioning
=
0.
We
denote the set of
also the relevant resource constraints (which
and suppress
useful,
is
/
Throughout, we condition strategies on pubhc history
be specified below) by Mt-
will
and
will lead to
this
/i'
dependence when the context makes
the conditioning clear.
The
models with no commitment
typical assumption in
is
that the mechanism designer
can commit to a submechanism at a given date, but cannot commit to what mechanisms
will
be offered
in the future.
government whereby
same
period.
following
The
it
In our context, there
can use
an additional type of deviation
for the
power to extract resources from the society even within the
its
interaction between the government and the individuals
game form
modeled with the
is
at each date:
1.
At the beginning
2.
Individuals send a message
h^"^ 6
is
of period
t,
the government offers a submechanism
e
zt
Zt,
and
G Ht
ht
is
Mt E M.
and
realized; zt
H^^^ and z*~^ G Z*~^ determine labor supplies according
ht together
to the
with
submechanism
Mt.
3.
Production takes place according to the labor supplies of the individuals, with Yt
F[Kt,Lt
Lt
(/r*)
(/'*)),
=
/q
k
where Kt
(2''',/i*) di,
The government
it (^')
£
where
pre-specified
=
it
will
all
become
indexes individuals and
production
0,
=F
=
1,
function
c't
If ^j (ft*)
new consumption
(ft*)
[0, 1]
{Kt, Lt
(ft.*))
-
E Z* denotes the
z^'^
is
distributed
submechanism M^, denoted by
among
submechanism Mt £ A4t, the government chooses
(z''*,ft') di.
Jq Ct
K^+i
G
i.
and next period's capital stock
a
i
decides whether to deviate from the
{0; !}• If it {^^)
=
the capital stock inherited from the previous period, and
is
history of reports by individual
4.
{h})
determined as Kt+i
is
x't
(ft.*)
X( (/)*)
=F
the government chooses x[
:
agents according to the
< F
[Kt,Lt
(ft*)
< F
(^Kt,
(ft'))
—
Lt
(/i*)),
it (ft*)
(A't,Lt (ft*)),
and
Z^ x H^ -^ R+, and next period's capital stock
(ft*)
-
/o
cj (s'-*, ft*)
clear below, the government's strategies allow
di for all
ft.*
e
—
is:
if*.!^
submechanisms to be functions of the reports of
agents in the past.
Finally,
we could
at that date to
define a
submechanism as a mapping Mt
emphasize that what can be achieved
will
[Kt] conditional
on the capital stock of the economy
be a function of the capital stock.
We
suppress this
dependence to simplify notation.
'^More generally, we can allow the government to capture a fraction 77 < 1 of the total output of the
economy when ^ = 1, where the level of rj could be related to the institutional controls on government
or politician behavior.
In this case, the constraint on the government following a deviation would be
K't+i {h') = r}F[Ki,Lt (/i')) — x't [h') — /q cJ [z^'^ ,h}) di, with the remaining 1 - 77 fraction of the output
10
This game form emphasizes that the only difference between the standard models with no
commitment and our setup
is
that the government, in the last stage, can also decide to expro-
priate the output produced in the economy. Notice that at this stage, labor supply decisions
have already been made according to the pre-specified submechanism Mt. However, consumption allocations cannot be
of the output for
M—
Let
by M..
{-A<ff}^o
=
Let x
sumption
Cj
Z* x if ' -+
:
M+
let
a mechanism, with the set of mechanisms denoted
which
/i*,
by
=
Pi
lMt,^i
conditional on history of publicly
t
,xt [h^] ,Xj
(/i*)
(/i*) ,5^ j.
the submechanism that the government offers at stage
=
Mj
Since
0.
element, Xt,
function
is
is
is
levels
6
if
time
satisfy: xt
< F
her type history
be the
set of p^'s
this
is 0*,
is
and
p*
< F
e
^j ==
L
getting destroyed.
The
fourth
is
the
capital stock for the following period,
\
pj's
up
her history of messages so far
effect
up
to
and including
is
at time
and the publicly observed
z*~^
to time
i
t
—
1
on our results and
are p*~-^ and h^~^.
for
now, we set
77
=
return to issues of institutional limits on government expropriation below.
we
are characterizing a (sustainable) mechanism, the ownership of the capital stock
Instead, this
is
government decides how
simply the amount of resources used
this production will
'^In fact, p' includes the action
appropriately, only a subset of p'
for
it,
-^^
z^~^, p*~^,h^~^) as the action of individual
This generalization has no
we write
{Kt, Lt), but to simplify notation
publicly observable.
{9^
Government consumption
We
consequence
citizens,
Finally, the fifth element
denote the history of
TV'
of government actions and aggregate variables
Since
when
Once again the
{Kt,Lt) and x[
For the citizens, define a\
^
the
what the government consumes
determined as a residual from the resource constraint.
and assume that
specified.
is
determined as a residual from the resource
is
the government consumption level
IR+. Let Tit
notation.
and the second
pj
that the government chooses after deviating from the original submechanism, with
c^
must
i,
is
t,
element of
first
not specified as part of the action profile of the government. -^^
Ct denoting the set of all such functions.
Kl_^i,
Pf.
The
both total production and total consumption by the
specifies
given xt the capital stock for next period, Kt+i,
constraint and
of time
1
government's expropriation decision. The third element of
<^j
an implicitly-agreed sequence of
is
government.
levels for the
represent the action of the government at time
itself if
new consumption
the government choose a
define a social plan as {M,x),
observable variables,
is
expropriating some
is
at this point.
Mt G A4t be
with
submechanisms and consumption
We
Mj, since the government
{xf (/i')}^Q be the (potentially stochastic) sequence of government con-
We
levels.
to
Consequently, we also
itself.
allocation function,
made according
in
production
when
histories
The
1
t
action
to simplify
Kt+i is not
and the
in the following period,
be distributed.
and the function cj, which are not observed when ^, =
should be observed publicly. This slight abuse of notation
x't
the analysis.
11
0.
is
Thus, more
without any
aj specifies a message
zt
£ Zt,
so:
aj
We
write
type
a*
A
histories h^~^.
=
z^~\p^-\h^-^)
(0*
Z*'i X 71*"^ X 0* X if*"!
^
Zt.
to denote the message resulting from strategy at for an agent of
z' (at (0*,/i'~-'))
and given public
6*'
:
strategy
truth telling
if it satisfies
7^*-l
and h^'^ G if*"\
e 6*, z*^^ G Z*-\ p*"^ G
for all 0*
zt [e^]
is
I
(3)
where the notation
true
Qf
type.'^'^
{^9t
means that the individual
zt [O^]
To economize on
z''~^ [^'
''l
I
iP'
=
Q^*-
sending a message that fully reveals her
we represent the
notation,
\^*~^)
is
by
truth-telling strategy
Notice that this strategy only imposes truth-telling
lowing truthful reports in the past (si"re instead of an arbitrary history of messages
have conditioned on z^~^
a®
{9t
[0*~'^]).
z'-\p'-\h'-')
-
I
where zf stands
for a
In addition,
let
zf for all 9' G 9*, z'-' G
p'-' G 7^*-l
Z*-\
message signifying that the individual
Assumption 3?^ Therefore, the individual can always choose
disabled
is
is
and h''^ G H'-\
(i.e.,
91
=
^o)-
Mt
an element of
Such
because of
to supply zero labor, or in other
words, any feasible mechanism (submechanism) must allow for "freedom of labor supply".
notation a]
{9t
\
z'"^, /0*~\ h^~^)
null strategy. Finally,
we denote the
A
all
denoting the set of
=
a^ to denote that the individual
We
playing the
is
by a, with
strategy profile of all the individuals in society
such strategy
we
z''~^,
us define the null strategy
a message must always be allowed in any submechanism that
will use the
fol-
profiles.
t
Let
Z(
G Zt be a
government's strategy at time
determines
Mt G Mt,
Ct
a function of the government's
We
usual,
we define
—
Z'-
W
Zg.
The
:
7l*-i X
e {0,1},
own
Xt
Z*-i X i7*-i -.
G
[0,
F (A't, Lj)],
and q G
Ct as
7^,
x't
G [0,F
(ATjjLt)]
past actions and the entire history of reports by citizens.
denote the strategy of the government by F and the set of government strategies by Q.
^"Whenever we write
may have
^'
As
therefore
t is
Yt
i.e., it
time t?"
profile of reports at
"for all /i'~'
G /f'~\"
this should
bo understood as "almost surely," since some histories
zero probability.
Since an individual with
6\
—
6o cannot supply
any labor, he must always send the message
an
z^
(or
Z(,
where
equivalent message).
i
^^More formally, 3j assigns a report to each individual, thus it is a function of the form
G [0, 1] denotes individual i, and Zt is the set of all such functions.
12
z
:
[0, 1]
—
.
Definition
citizens
1
A
f Perfect
Bayesianj equilibrium in the game between the government and the
given by strategy profiles
is
each other in
F and a
information sets given
all
Bayesian updating given the strategy
=
\
Mt,it
and whenever
beliefs,
We
profiles.'^^
profiles are best responses to each other as
Let us define Tm,x
that are sequentially rational,
a for
M
a sustainable
is
T ^a T for
and a
the citizens
strategy profile
librium and induce an action profile
such that
for
all
Mt — Mt
T ^
sustainable
Q.
,
£,t
(/i*)
=
0,
all
(^*) i^« (^') i^t {^^)
mecbanism
and
if
possible, beliefs are derived
from
write the requirement that these strategy
action profile of the government induced by strategy
Definition 2
best responses to
i.e.,
^t
i
T
T £ Q and a
^
f
a
>:p
for
all
a£ A.
^^^ (potentially stochastic)
given a social plan (M, x).
there exists x
=
{xt
(/i*)
},_q,
o,
strategy profile
Tm,x G Q for the government, which constitute an
[Mt,^i{h'),it{h'),xi{h'),c't}^^^
xt {h})
—
xt
(/i*)
for
all h*
equi-
for the government
G i^S and
Fm^x ha F
satisfies
In this case, we say that equilibrium strategy profiles Tm,x o.nd
a support
the
M
mechanism
In essence, this implies that the government does not wish to deviate from the social plan
(M, x) given the strategy
profile, a, of the citizens.
The notation F ^„ F makes
this explicit,
stating that given the strategy profile, a, of the citizens, the government weakly prefers
strategy profile to any other strategy profile based on the
same
its
implicit agreement.
Truthful Revelation Along the Equilibrium Path
3.2
The
revelation principle
is
a powerful tool for the analysis of
Winston and Green,
tation problems (see, e.g., MasCollel,
the government,
who
mechanism design and implemen-
1995). Since, in this environment,
operates the mechanism, cannot commit and has different interests than
those of the agents, the simplest version of the revelation principle does not hold; there will
exist situations in
which individuals
will prefer
not to report their true type
1984, Preixas, Guesnerie and Tirole, 1985, or Bisin and Rampini, 2005).^^
this section will
first
result of
all
long as the set of sustainable mechanisms
do not introduce
(i.e.,
explicit notation to describe beliefs, since these
will
positive discount factors).
consider the problem of finding the best allocation for individuals.
will see below, as
We
The key
be that along the equilibrium path, a version of the revelation principle
hold (without introducing a fictional mechanism designer and for
Let us
Roberts,
(e.g.,
the constraint
do not play any
As we
set, (5)-(7)) is
role in
any of the
analysis or the proofs.
^As noted
It is
in the Introduction, this
statement refers to the case in which messages are sent to the government.
possible to construct alternative environments with fictional
power, so that the revelation principle holds.
13
mechanism designers with
full
commitment
.
nonempty,
this
equivalent to choosing the best sustainable mechanism, given by the following
is
program:
oo
maxE ^^*u
(z* [a*
i^ct
{e\ h'-') ,h']), k {/ [qj {e\ h'-') ,h'])
(4)
6]
\
subject to the resource constraint,
= F{Kt{h'''),jlt{z'[a^{e\h'-')]M)dG'{6')
Kt+i{h')
(5)
a set of incentive compatibility constraints for individuals,
a
is
and the "sustainabihty" constraint
E Y,5'v{x,^s{h'^^))
h'
a best response to Tm,x-,
(6)
of the government:
>
max
E {v{x[)+5v^[K[^,rc't\M')}\h'],
(7)
s=0
for
alH >
and
^ H^. Note that current labor supply decisions are conditioned on the
all h*
reahzation of the public history up to time
the previous period,
The
Kt-,
will receive
the implicitly-agreed consumption schedule for
itself.
from date
t
onwards by sticking with
The right-hand
side
is
can receive by deviating. The potential deviations include a deviation at the
subgame
at time
individuals,
cj;
submechanism
=
^t
1,
to expropriation, ^i
t
=
or ^j
at time
—
and a choice of
xj
the government chooses Xj, K[j^i and
by current
utility,
v{xt),
together with a
\,
(/i*)
different
and continuation
c[
to
from Xt
maximize
its
maximum
the
(/i*);
or the offer of a
v1).
\
for
new
In the case where
deviation value, which
value, written as if IK[_^_i,c[
it
last stage of the
new consumption schedule
(encapsulated into the continuation value
i -|- 1
inherited from
the possible deviations by the government at date
all
what the governinent
is
while the capital stock,
h*'~^
encompasses
last constraint, (7),
the left-hand side
t:
conditioned on
is
/i',
t,
MM,
is
given
to emphasize
that this continuation value depends on the entire history of submechanisms (thus information)
up
to time
t,
M*, and on the
this constraint, (7),
some sequence
of
satisfied.
The
satisfied,
it is
either because the
submechanisms or consumption
government prefers
is
were not
capital stock from then on, K^^-^, as well as potentially
^^
—
latter,
1.
i.e.,
In the former case,
^^
=
1,
government prefers
levels different
from
(Af, x), or
we can always change (M,
^^
on
=
c[.
If
and
because the
x) to ensure that (7)
cannot be part of the best equilibrium allocation from the
14
.
viewpoint of the citizens, since
involves government expropriation. Consequently, as long as
it
the constraint set given by (5)- (7)
is
nonempty, the best allocation must
and
satisfy (7)
is
thus
a solution to the program of maximizing (4) subject to (5)-(7). Finally, this constraint set
is
indeed nonempty, since the trivial allocation with zero production and zero consumption for
parties
all
is
in the set.^^
Let us also introduce the notation
viduals play
Lemma
a along
1 Suppose
a— {a
\
denote a strategy profile where
a') to
We
the equilibrium path and a' off the equilibrium path.
Assumptions 1-5
E Y,s^v{x,+4h'^^))
Then
hold.
>v{F
h'
all indi-
then have:
any sustainable mechanism,
in
{Kt ih^^^)
,
Lt
(/i*)))
for all
and
t
/i*
H\
e
(8)
s=0
is
Moreover, in the best sustainable mechanism,
necessary.
M*
e
yVf*, K[j^j
e
M+ and
Cj
6
and the
Ct,
v^ iK^_^j,c[
sustainability constraint (7)
is
\
M*) —
for
all
equivalent to (8).
Proof. See Appendix B.
This
lemma
of capital stock
uses the fact that irrespective of the history of
future production, there
left for
is
submechanisms and the amount
an equilibrium continuation play that gives
the government zero utility from that point onwards (which
is
analogous to the results in
repeated games where the most severe punishments against deviations are optimal,
1988).
This continuation play
Abreu,
used as the threat against government deviation from the
is
The
implicitly- agreed social plan.
e.g.,
implication
is
the best deviation for the government involves
<^t
that, along the best sustainable
=
1
and
x[
= F {Kt, Lt).
mechanism,
This enables us to
simplify the sustainability constraints of the government to (8), which also has the virtue of
not depending on the history of submechanisms up to that point. ^^ Moreover, the
shows that
Remai'k
in
1
Assumption
Lemma
1.
1
any sustainable mechanism
By
allowing
all
citizens to claim to
3 plays a crucial role in the proof of
without Assumption 3
At the beginning
of period
is
i,
be the worst type in the punishment phase,
Lemma
.
the government offers a
^^In particular, the following social plan
all
1
An
alternative
game form
delivering
as follows
possibly dependent on past histories, denoted by
labor (and consumption) to
also
necessary.
is
(8)
lemma
{M,x)
is
menu
of labor-consumption bundles,
M.f
a sustainable: xt
=
for all
t
and Mt always
Eissigns zero
reports.
^^This statement refers to the sustainability constraint,
cations depend on the history of individual messages.
15
(8).
The optimal mechanism
will clearly
make
allo-
.
2.
Individuals choose
3.
The government
how much
labor to supply, k 6
decides ^^ G {0, 1}.
or distributed as before.
chosen labor supply
=
If ^(
If ^j
=
menu
not in the
is
and production takes
,
is
distributed
is
among
those
agents
who have
who have chosen a
M+
such that $ct €
(i.e., It
place.
expropriated, and consumed
menu -Mp, and
level as specified in the
labor supply level that
^
output
1, all
production
0,
[O,
with
[lt,Ct)
G
Ct),
receive zero consumption.
This game form directly imposes "freedom of labor supply" meaning that
it is
,
who
decide
how much
labor to supply, whereas with our original
determines
how much
labor individuals supply.
without Assumption
reasons:
the
first,
the
3.
We
mechanism design
game form
chose the
game form
in the
With
main
text
the submechanism
game form. Lemma
this
main
1
applies exactly
and Assumption
3 for
two
closer to the standard Mirrlees setup
and
in the
is
game form,
individuals
text
game form described here imposes an
structure; second, the alternative
additional restriction on the set of mechanisms, such that individual allocations can differ only
to the extent that individuals have chosen different labor-consumption bundles in the past.
In contrast, our more general
individuals
bundle
who have made
in the past
results,
may be
game form
allows history-dependent allocations in which two
different reports
but have received the same consumption-labor
treated differently in the future. Since they both lead to identical
whether our baseline game form with Assumption 3 or
without this assumption
is
preferred
is
this alternative
game form
a matter of taste, with no consequence for the rest of
the results in the paper.
Remark
2 Yet another possible
game
form, which would lead to similar results, gives citizens
the option to replace the government after a deviation.
obtain a similar result to
Lemma
1.
replacement
If
is
costless,
we would
Acemoglu (2005b), Acemoglu, Golosov
See, for example,
and Tsyvinski (2006).
M*
Next, we define a direct (suh)mechanism as
direct
mechanisms involve a
their current type.^^
We
restricted
:
6* x
message space, Zt
—
i?*
=
>
[O,
/]
x M+, In other words,
Qt, where individuals only report
denote a strategy by the government inducing direct submechanisms
along the equilibrium path by V*
Definition 3
we have
A
strategy profile for the citizens, a*,
that a] {6^
|
9^~^,p^''^, /i*~^)
=
a*
.
We
write
is
truthful
a*—
{a*
\
if,
along the equilibrium path,
a') to
denote a truthful strategy
profile.
"^Once again, more generally,
mechanisms.
this can
be written as A/*
16
:
©' x
//'
— A
>
([O,
J]
x
R+)
to allow for stochastic
The notation a*=
a')
emphasizes that individuals play truth-telling along the equilib-
rium path, but may play some
different strategy profile, a', off the equilibrium path. Clearly, a
(a*
\
truthful strategy against a direct
agent.
mechanism simply amounts
Let us next define c[T,a,h], l[T,a,h] and x[r,a,h]
to reporting the true type of the
the equilibrium
as, respectively,
consumption and labor supply distributions across individuals
a function of the history
(as
of their reports), the sequence of government consumption levels resulting from the strategy
profiles of the
that
all
of time
government and individuals, and the sequence of
of these functions only condition or information available
1
=
to time
t
{/i*}j^Q,
such
for allocations
(Truthful Revelation Along the Equilibrium Path) Suppose Assumptions
1-5 hold and that
for
up
h
t.
Theorem
anism.
histories
F and a
are a combination of strategy profiles that support a sustainable mech-
Then, there exists another pair of equilibrium strategy profiles T* and a*
some
a'
such that T* induces direct submechanisms and a* induces truth
equilibrium path, and moreover c[T,a,h]
—
c[T* ,a* ,h\, l[T,a,h]
=
=
[a*
\
a')
telling along the
l[T*,a*,h] and x[T,a,h]
—
x[T*,a*,h].
Proof. See Appendix B.
The most important
implication of this theorem
is
that for the rest of the analysis,
mechanisms on the
restrict attention to truth-telling (direct)
side of the agents.
we can
The reason
why, despite the lack of commitment and the self-interested preferences of the mechanism designer, a revelation principle type result holds
is
twofold:
first,
the government has a deviation
within the same period; and second, individuals can use punishment strategies involving zero
labor supply following a deviation by the government.
support a sustainable mechanism, making
it
The punishments
strategies of citizens
the best response for the government to pursue the
commitment on
implicitly-agreed social plair (M, x). Given this sustainability, there
is
effective
the side of the government along the equilibrium path. This notion
is
important to distinguish
from the commitment that
exists in the
unconditional commitment
(i.e.,
no commitment
along
standard mechanism design problems where there
is
In contrast, in our environment, there
is
where the government can exploit the information
it
all
off the equilibrium path,
paths).
has gathered or expropriate part of the output. In
reporting by the individuals
is
fact, off
the equilibrium path, non-truthful
important to ensure sustainability. Nevertheless, by definition,
along the equilibrium path induced by a sustainable mechanism, the government prefers not
to deviate fi'om the implicitly-agreed social plan and thus individuals can report their types
without the fear that
this information or their labor
17
supply
will
be misused.
.
3.3
The Best Sustainable Mechanism
Theorem
enables us to focus on direct mechanisms and truth-telUng strategy a* by
1
individuals. This implies that the best sustainable
mechanism (and thus the best
all
allocation)
can be achieved by individuals simply reporting their types. Recall that at every date, there
is
This implies that
an invariant distribution of 9 denoted by G{9).
which
distribution,
is
simply the t-fold version of
G [9),
individuals, each history 9^ occurs infinitely often). ^®
total labor supply given history
Cf
(/i*)
=
ct (0*,
Jqj
mechanism
h*)
dC*
as Lt
/i*
(0*).^^
=
/qj k {9^
{9) (since
Given
,
/i')
mechanism with
1
there
has an invariant
is
a continuum of
this construction,
dG^
Moreover, since Theorem
equivalent to a direct
is
[h*')
G^
9^
[9^)
,
and
total
we can write
consumption as
establishes that any sustainable
on the side of the agents, we
truth-telling
obtain the main result from the section, which will be used in the rest of the paper:
Proposition
Suppose Assumptions 1-5
1
hold.
Then, the best sustainable mechanism
is
a
solution to the following maximization program:
jjSM_
max
IE
{ct{e',k')M{e\h*),xtm,i<t+im}Zo
some
subject to
Kt+i
(/i*)
initial condition
=F
Ut
{h'-')
,
J2P'u{ct{{9^'\h')),k{9''\h')\9l)
(9)
t=o
Kq, the resource constraint
J
It
{9\ h') dG' [9')\
- Jct
{9\h') dG'
[9')
- xt
{h')
,
(10)
a set of incentive compatibility constraints for individuals,
E Y^p^u
(ct+, (0'•*+^/l'+^) ,k+, (0'*+^/^*+^)
1
T'Kh^
91^,
(11)
oo
>
E
f^Kh^
s=0
^f_i_^ \
,
and the sustainability
constraint of the government
E J2S'v{x,+4h'+^))
h'
>
V
(f
(^Kt
{h''')Jlt {9\h') dG' (e*)))
,
(12)
.s=0
for
all t
and
h^
£ H^
^^More formally, given the continuum of agents, we can apply a law of large numbers type argument, and
&' will have positive measure. See, for example, Uhlig (1996).
^^From now on, we suppress the "'s to simplify notation and simply use ct, It and Xt- Note also that /q, here
denotes Lebesgue integrals, and in what follows, we will suppress the range of integration, 0'.
each history
18
Proof. The proof follows from
rium (a**,r**), that maximizes
<^j
=
1 for
any
t.
and
h*
e
(9).
By
and Theorem
1
1.
Suppose there
an equilib-
the argument in the text, (a**,r**) will not feature
\Mt>
government, {xt {h^)}^^Q and
^j (/i*)
HK
consumption
levels for the
Then
(M,x)
setting
=
sustainable direct mechanism. This direct
1
mechanism has
be written as (11) since T* induces direct mechanisms.
Finally,
important to note that the objective function
expectation at time
t
~
before individuals
know
=
for
to satisfy the resource constraint,
from
which instead of
Lemma
1,
(6)
can
the constraint
m
a best response to citizens' strategies, a*,
is
(po-
to find (a*,r*) corresponding to a
(10), the incentive compatibility constraints of individuals at all dates,
(12) ensures that T*
,
,{it{h^)}Zo) implies that (a**,r**)
({^^t}°°
support a sustainable mechanism. Then, use Theorem
It is
exists
Therefore, {a**,T**) features a sequence of submechanisms
tentially stochastic)
all t
Lemma
in the optimization only incorporates
their type
(i.e.,
"behind the
veil of igno-
rance"), while the incentive compatibility constraints, (11), require that there are no profitable deviations given
any sequence of individual, types and public
that the maximization in (9)
is
histories.
Note further
over sequences [ct {0^,h^) Jt {&^,h^) .^t {h^) iKt+i (^*)}^n'
which allows potential randomization conditional on the realizations of the publicly observable
aggregate variable
sustainable
The
program
ht-
Finally, this
mechanism
role of
also defines
U"^^
as the ex ante value of the best
an individual.
for
Theorem
problem
in this formulation
1
mechanism
for the best sustainable
is
obvious, since
as a direct
it
enables us to write the
mechanism with
truth-telling, thus
reducing the larger set of incentive compatibility constraints of individuals to
Sustainable Mechanisms and the Quasi-Mirrlees
4
(11).'^°
Program
Let us next define the dynamic Mirrlees program (with full-commitment, benevolent gov-
ernment and exogenous government expenditures). Imagine the economy needs to finance a
(potentially stochastic) exogenous
dynamic Mirrlees program
tive agent,
of
can be written as
government expenditure Xt
maximizing the time
(e.g.,
t
=
(/i')
>
at time
t.
Then
the
{ex ante) utility of a representa-
Golosov, Kocherlakota and Tsyvinski, 2003, Kocherlakota,
2005):
oo
max
E
.f=0
^''The equations in (11) focus on the 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
any sequence of reports
covered by this set of constraints.
19
<
Otj^^ \
i
thus any potential deviation from
+ Xt (/i*) + A't+i (h}) <
= J Ct{9\h^) dG {9^) and Lt{h^) -
subject to the incentive compatibility constraints, (11), and Ct (^*)
F[Kt{h^-^),Lt{h^))
/
It
(0*, /i*)
dG
[6*-)
we can add the
,
e
for all h^
and
H\
{9\ h^) and
ct
where Ct{h^)
/( [9'-, /i*)
feasibility constraint that
{Ct
(/!*)
,
are jFj-measurable (see footnote 12). Moreover,
[Xt (/i')}^o should be such that
Lt (/i*)}^o e A°° for
all
G
/i*
H\
where
A°°
=
{Ct,Lt]r=a such that 3
{
=
Ct {h')
{c* {9') ,lt
fct {9\h') dG{e')
,
(0*)}"o satisfying
and Lt
=
{h')
(13)
(11),
f h {9\h') dC
{9')}.
In other words, for certain government expenditure sequences, {Xtj^^o's, the constraint set of
maximization problem can be empty
this Mirrlees
(e.g., if
Cf
,
Lt (/i*)}^o G ^°° for
For a sequence {Ct
program
(/i*)
,
all /i'
6
and Lt
Thus
compatibility constraints of individuals cannot be satisfied).
that {Ct (h*)
=
it is
>
0,
the incentive
important to ensure
HK
Lt (/i*)}t^o ^ ^°° ^^^
^11
/i*
e
i?',
we can
define the quasi- Mirrlees
as
oo
E
max
W({Ct(/i*),L,(h*)}~o)^
{c,(9',^').''('^'.''')}::o
;/3*t.(Q(0'-*,/i*),/t(0''*,/i*)i0j)
(14)
t=o
subject to the incentive compatibility constraints, (11), and two additional constraints
fct{eKh')dG{9')<Ct{h'),
(15)
jlt{9\h')dG{9')>Lt{h'),
(16)
and
for all
/i*
€ H^-
Clearly this program takes the sequence {Ct (h^)
maximizes the ex ante expected
ibility constraints as well as
consumption
utility of
Lt.
The
levels across agents for all report histories to
functional
U{{Ct
(/i')
sum
show
in
,Lt (^*)}j^o) defines the
the Appendix that W({Ct,Lt}Xo)
first,
(15), requires the
is
sum
of
be no greater than some number
of labor supplies to be no less than
agent in this economy for a given sequence {Ct {h}) ,Lt
'''We
and
an individual subject to the usual incentive compat-
two additional constraints. The
Ct, while the second, (16), requires the
Lt (^*)}^o ^s given
,
maximum
ex ante
[t
=
some amount
0) utility of
an
(/i')}(^q.'^^
a well-defined functional.
Nevertheless, the incentive
embedded in (11) do not form a convex set. For this reason, in the Appendix, we follow
Prcscott and Townsend (1984a,b) and allow lotteries to convexify the constraint set and establish concavity of
U{{Ct, Lt}Zo) in {Ct, L(,}^o- This will change the exact form of the optimization problem, but not its economic
essence. For this reason, we relegate the formalism of the lotteries to Appendices C and D, and in the text, we
assume that U{{Ct,Lt}tZo) is concave. In the Appendix, we also prove that U{{Ct,Lt}^o) is difi^erentiable,
which we again assume to be the case in the text.
compatibility constraints
20
Returning to the dynamic Mirrlees program,
a given sequence of government expendi-
for
tures {Xt (^*)}^0' ^^^^ ^^^ clearly be written as:
max
subject to a given level of
Kq and
Ct ih')+Xt {h')+Kt+i {h')
Therefore,
to
<F
[Kt {h'-'),Lt {h'))
,
A~,
(18)
as a solution to a two-step
max-
{Q
and
we can represent the dynamic Mirrlees program
imization problem, in which the
tional
(17)
{h'),Lt (/i*)}~o e
HK
G
for all h^
U{{Ct{h'),Lt{h')}Zo)
W({Ct
(/i*) ,1/t
first
step
is
the quasi-Mirrlees formulation, yielding the func-
(^^If^o)' and the second step
over (potentially stochastic) sequences {Ct
is
the maximization of W({C(
(/i*) ,-Lf (/i*)
,Kt+i
(/i*)}
(/i*)
,Lt {h^)}tZo)
subject to a resource con-
strained and feasibility.
Now
again using the quasi-Mirrlees formulation, the characterization of the best sustainable
mechanism,
(9),
can be wrritten as
subject to
Ct [h')
+ Xt
(/i')
+ Kt+i
for all h*
G H^, and
program
in (17)-(18)
{h')
< F
{Kt [h}'^),Lt {h')) and [Ct {h'),Lt {h')}Zo e A°°,
also subject to (12).
The only
difference
and the best sustainable mechanism
of the sustainability constraint for the government, (12),
(20)
between the dynartiic Mirrlees
in (19)-(20)-(12)
which
also
the presence
is
makes {xt (/i*)}^o
endogenously chosen sequence instead of the exogenously given {Xf}^g.
^^^^
This formulation
establishes the following theorem.
Theorem
Then, the best sustainable mechanism solves a
2 Suppose Assumptions 1-5 hold.
quasi-Mirrlees program for
some sequence {Ct
[h^)
Lt (/i')}^o
,
Proof. This follows immediately from rewriting
maximization program, and expressing
Kt+i{h').
(10) as Xt
(9)- (12)
(ft*)
= F
^ ^'^
from Proposition
{^Kt (ft*"'')
,Lt
1
(ft*))
—
Ct (h^)
—
m
Consequently, any allocation consistent with the best sustainable mechanism
to a
as a two-step
problem that maximizes the ex ante
utility of the citizens.
is
a solution
Despite the political economy
constraints and the resources extracted by the government from the society, the mechanisni
21
maximize the ex ante
will
utility of the citizens given
imposed by
in addition to the resource constraints
some resource constraints (which
feasibihty).'^^
are
This theorem also enables
us to represent the difTerences between the dynamic Mirrlees program and the best sustainable
mechanism purely
{Ct {h})
,
program
in
terms of aggregate distortions, corresponding to what the sequences
Lt (^*)}j^o S ^°° ^^^
['^Tvd
how they
differ
from the solution to the dynamic Mirrlees
in (17)- (18)).
To make more
progress,
we need
to characterize the behavior of the sequences {Ct, if }^o
(and {xt}^Q) under the best sustainable mechanism, which
The Economy with Private
5
The dynamic behavior
is
what we turn to next.
Histories
of the optimal sustainable
mechanism
is
determined by the need to
provide dynamic incentives both to the government and to individual agents. As
(e.g.
is
well
known
Green, 1987, Phelan and Townsend, 1991, or Atkeson and Lucas, 1992), the behavior of
individual allocations can be very complicated even in the absence of sustainability constraints
on the government. In order to highlight the
we
first
effect of
consider mechanisms with private histories,
government sustainability constraints,
i.e.,
where individual
observed by the government. '^^ The restriction to private histories
is
histories are not
purely a heuristic device,
useful in separating different parts of the analysis. Section 6 below wih drop this assumption
and characterize the best sustainable mechanism
restrictions
^"It
is
on potential mechanisms or
also interesting to highlight
in the history-dependent case without
any
strategies.
which features of our environmont are important
for
Theorem
2.
For this
purpose, consider an environment without capital and suppose that labor supply is equal to output. Suppose
that the government can tax individuals differentially according to how much they produce, with the meiximum
amount that can be extracted from an individual supplying labor I as fj{l), where r} \0,l\ -^ [0,l\- In this
:
case, using
an analog of
Lemma
1
and suppressing dependence on public
histories,
we have the
sustainability
constraint as:
f^S%{xt+s)>v(ffi{l{e'))dG'{9')Y
^-^
s=0
where
the
/
(6*')
is
^
the labor supply of an individual with type history 9\ and the term
maximum amount
/
i)
(i (61'))
that the government can expropriate given the technological restriction
dG^
[0*]
captures
embedded in the
Theorem 2 does
function ^(•) and the distribution of types given by G' (5*). Unless 77(-) is a hnear function.
not apply, and there would be further distortions relative to our baseline analysis.
^^Here the term "private histories" refers to the fact that past reports of individuals are not conditioned upon,
and
is
not related to the public histories,
h*'s,
which designate the potential randomness of aggregate variables.
22
Best Sustainable Mechanism with Private Histories
5.1
To
further simplify the notation in this case, let us also assume that there
the aggregate production function of the economy
Yt
where Kq
The
—
is
(21)
imphes that
time
at
in admissible
environment the incentive compatibility constraints
and written
u
for all 9t
in
&
Q
and
Assumption
9t
t.
mechanisms, allocations must
{ct
{9u h')
& Q, and
3, (22)
,
k (Bu
for all
h')
t
and
A''
+
now
h*'
1
/i*).
In such an
can be separated across time
(q. (Ot, h')
,
k
[h /^*)
E H^ Moreover, given the
.
I
(22)
^t)
single crossing property
set of incentive compatibility constraints only for
types in 0, this implies that (22)
The
is
equivalent to TV
mechanism with private
best sustainable
+ xt
(/i*)
<
Lt {h')
the quasi- Mirrlees program defined above.
(23)
,
It is
straightforward to see that
because of "private histories", the optimal allocations of {ctJt) depend only on Ct
Lt [h^) and are independent of any Cg (h^), Lg
is
histories
subject to (12), (22) and the resource constraint
Ct [h')
Recall
\et)>u
can be reduced to a
incentive compatibility constraints."^^
(9)
for agents
histories,
as
neighboring types. Since there are
maximizes
capital, so that
= Lu
depend only on agents' current report (and potentially on public
periods,
no
simply
and Lt denotes the aggregate labor supply
restriction to private histories
is
time separable,
i.e.,
U{{Ct
valued differentiable function
{h^) ,Lt
1/
:
M^.
(h'^)
s
The program
and
^t. This implies that U{{Ct, Lt}^^)
= ^ EZoP^^i^t
{h^)}Zo)
^ K.
with
(/i*)
{h*),Lt (h^)) for some real-
for the best sustainable
mechanism,
(19)-(20), therefore becomes:
max
M,
>
/3*C/(Ct(/i*),Lt(/i*))
(24)
{Ct{hi),Lt{h')Mh')}^o
subject to the resource constraint, (23), and the sustainability constraint.
E
J2S'v{^t+sih'+'))\h'
>viLt{h*)),
(25)
,s=0
N
specifically, in pure strategy direct mechanisms, there will he
{N + 1) incentive compatibility conand Assumption 2 makes sure that only TV of those, i.e., those between neighboring types, where the
higher type may want to misreport to be the next lower type, may be binding.
More
straints,
23
for all
As
{C,L)
t
and
HK
e
h^
problem
before, this
which
pairs,
is
and (22) are
(15), (16)
Assumption 6
such that
well defined only for
is
some {C,L).
a simplified version of (13), by A,
Also define
satisfied}.
w=
satisfied
iD, is
if
A = {iC,L) B{c{9) ,l{0)) s.t.
{L — C) / {I — 5). We impose:
^^
6 argmax(c'_^)£yv^
(-^
utility that
C*)
/ (1
—
5),
can be given to the gov-
sufficient to satisfy its sustainability constraint (25). Clearly this
the discount factor of the government,
We now introduce our key
~
v{L).
This assumption ensures that the highest discounted
ernment,
denote the set of such
:
max(c,i,)eA
(sustainability) There exists {C,L)
v{L-C) / (1-5) >
i.e.,
We
6, is
assumption
is
sufficiently large.
concept of '^aggregate distortions''. Since our objective
is
to
com-
pare the additional distortions created by the self-interested and time-inconsistent behavior of
the government, aggregate distortions are defined relative to the dynamic Mirrlees allocations.
Definition 4 In the model with no capital and with private
sequence {Cj
tially stochastic)
nism r*
for
is
all h*
it is
undistorted at
e H^
We
.
t'
is
say that {Ct
a natural definition.
mechanism coincide with the
(/i')
}^Q
It
be undistorted.
t
=
0,
t
[
^q
is
mecha-
a solution to (17) subject to (18) with
is
asymptotically undistorted,
if
—^ oo
dynamic Mirrlees program where government
being paid to the government under the best sustainable
is
if
the ruler could commit to a sequence of consumption levels
then Definition 4 imphes that the resulting allocation would always
The question
is
(/i*)
,Lt [h^) ,xt {h'^)}^^Q
is
whether when the sequence [xt {h'^)}^^Q
satisfy the sustainability constraint of the
Definition 4
,Xf (^*)}j^q induced by the best sustainable
allocations in the
can be noted that
at time
(/i*)
ue say that the (poten-
requires that the aggregate allocations in the best sustainable
It
expenditures are equal to what
mechanism. '^^
(/i*)
Ct (^*) ,Lt
if <
(almost surely) undistorted as
This
{xt
,Lt
(/i')
histories,
government,
(12), there will
is
determined to
be aggregate distortions.
general enough to cover the case in which (C, L) ^IntA (where IntA denotes
the interior of A). For the usual case where (C, L) £lntA, we have a simple condition characterizing undistorted allocations. In particular, since
when
(C, L)
GintA the solution
to the
U {C,L)
is
differentiable (see
Appendix B),
dynamic (full-commitment) Mirrlees program
(17)-(18)
satisfies:
Uc{aL) = -UL{C,L),
^^Note also that
if
an allocation
is
distorted, then in the absence of the sustainability constraints
modified to create a Pareto improvement.
24
(26)
it
could be
where Uc and Ul are the
(26)
intuitive:
is
partial derivatives of
and
C
with respect to
and
L. Condition
requires the marginal cost of increasing output by one unit to be equal
it
The next
to the marginal benefit of doing so.
distortions
U (C, L)
this condition
more
proposition illustrates the relationship between
explicitly (suppressing
dependence on
h^ to simplify nota-
tion):
Proposition 2 Suppose Assumptions 1-2
the marginal labor tax rate
1.
r^^f^t
=
the labor supply decision of the highest type of agent
is
the highest type of agent, 6 m,
o-t
time
t is
given by
+ UL{Ct,Lt)/Uc{Ct,Lt).
l
2.
on
Consider a sequence of {Ct, Lt}'^Q. Then:
hold.
if {Ct^,
it}^o
undistorted,
'^^
i.e.,
undistorted at
Uc
{ct
(On)
,
t,
k {Sn)
\
On)
=
-ui {a {9n) k {On)
,
\
On)-
Proof. Assumption 2 implies that we only need to check incentive compatibility constraints
for neighboring types.
Appendix C
establishes that
be the partial derivatives of u (which
Xi\ft is
at time
t,
vct
is
where the
Therefore,
(1
+ Aivt) =
i^ct,
Ul{ct{9N),h{9N)\0N){^
+ ^Nt) =
-l^Lt,
the multiplier on (15) at
= —UL [Ct, Lt)
Lt
1).
fi^iv)
I
exist.
first
U {C,L)
.
and the
t
and
definition of
vu
is
6^ and
the multiplier on (16) at
Lagrange multipliers, vet
=
t.
9j\f-i
By
the
Uc{Ct,Lt) and
Combining these equations, we have
ui{ct{eN).it{eN)\eN)
„
Uc[ct{QN)M{9N)\9N)
'^
.
^"^-'^
UhiCuU)
UciCt,Lt)'
equality defines r^^t, and the second equality establishes the
lemma. The second
Let Uc and ui
we have
the multiplier on incentive compatibility constraint between types
differentiability of
I'
by Assumption
exist
'"c(ct (6'7v),^t(6'7v)
where
Lagrange multipliers
result follows
immediately from setting C/i(C(,Lj)
==
first
part of the
—Uc{Ct,Lt) from
the definition of an undistorted sequence, in particular, equation (26).
Let us next follow
of the best sustainable
on
/i*),
(see
Thomas and Worrall
(1990) and consider the recursive characterization
mechanism. By standard arguments (and again suppressing dependence
any solution to the following recursive maximization problem
is
a sustainable
mechanism
Appendix C):
V{w)= max {U{C,L) + /3Viw')}
C,L,x,i
subject to
C-^x <L,
25
(27)
w=
v{x)
w'
where
w
is
v{x)
+
5w'
eW and
5w',
(28)
>
v{L),
(29)
[C, L)
e A.
the future utility promised to the government,
and the requirement that (C, L) G
A makes
sure that
consumption and labor supply. The program
level of
+
promised
in (27)
the set of feasible values for w,
is
at feasible levels of aggregate
determines optimal policies for a given
V (w)
the problem of finding the best sustainable mechanism
initial
W
we only look
w. Once the value function
utility
(30)
is
determined from this program,
is
simply equivalent to choosing the
value of promised utility to government, wq, such that wq 6 argmaxi,, V{w).^^
This program also makes the role of the sustainability constraint (29)
wishes to produce more output (or supply more labor L),
it
can only do so by providing
greater consumption to the government either today or in the future.
constraint
to aggregate distortions relative to the
The main
result of this section
=
1-3,
At
there
is
2.
Suppose that P
<
0,
ues {wt
(/i*)
is
dynamic Mirrlees benchmark.
the following theorem:
(/i')
an aggregate distortion.
5.
Let T* be the best sustainable
Then
}^_Q-
that Wt+i (/i*^^)
>
\^wt (/i*) } ,_„ is
wt [h*) for
all /i*+^
mechanism inducing a sequence
If
>
5,
of val-
a non- decreasing stochastic sequence in the sense
//'+^.
G
}t^n converges (almost surely) to
Moreover, a steady state exists in that
some w* €
[0, iv]
converges (almost surely) to some (C*,L*,x*), which
3.
this
5 and 6 hold.
1.
[wt
when
3 Consider the economy with no capital and with private histories and suppose that
Assumptions
t
Therefore,
binding, the social cost of increasing output will be greater than Ul, thus leading
is
Theorem
clear. If the society
is
and {Cf
(/i*)
,
Lt
(/i*)
xt (^*) }j^q
,
asymptotically undistorted.
then aggregate distortions do not disappear even asymptotically.
Proof. See Appendix C.
W
= [0, w] where u) is the
^''There is a number of technical details related to this program. First, we have that
maximal feasible and sustainable promised utility to the government, defined above. Moreover, in Appendix C
we sliow that there may be room for improving on this program by randomizing over the values of w and thus
over C and L (i.e., considering lotteries for the government in the same way as we do for individuals). We relegate
the discussion of this issue to Appendix C, but here we take the sequence of values given (promised) to the
government {tut}^o ^ ^ stochastic process, with each element taking values from the set W, and consequently,
the sequence {Ct, Lt}^^ that results from the best sustainable mechanism is also a stochastic sequence. Finally,
in the text, we assume that V is concave and differentiable, which are both proved in Appendix C.
,
26
Part
1 reflects
the additional aggregate distortions arising from the self-interested behavior
of the government. In particular,
a,t t
=
0,
there
is
necessarily a positive distortion, reducing
aggregate output. Intuitively, this distortion results from the fact that as output increases, the
more has
sustainability constraint (29) implies that
makes the
effective cost of
and
to be given to the government,
this
production higher for the agents. Consequently, the best sustainable
mechanism induces an aggregate
distortion, reducing the levels of aggregate labor supply
and
production below those that would arise in the dynamic Mirrlees program.
The most important
results are in parts 2
and
Part 2 states that as long as
3.
P <
6,
asymptotically the economy converges to an equilibrium where there are no aggregate distor-
and the marginal tax rate on the highest type
tions
in
combination with Theorem
by Mirrlees (1971)
economy
constraints
and
of the insights of the optimal taxation literature inspired
continue to hold.
will
equal to zero. Therefore, this theorem,
implies that despite the political
2,
commitment problems, many
the
is
Consequently,
when
the government
at least as
is
patient as the citizens, lessons from the optimal taxation literature are not only normative,
may
but
also help us
understand how tax systems are designed
are motivated by their
own
Part 3 of the theorem
is less
objectives, such as self-enrichment or reelection.
is
perhaps more important. This part states that
patient than the agents, distortions will not disappear. Since in
economy models, the government
the theorem
may imply
that in a
or pohticians are
number
Appendix
give a heuristic
B). Let 7
respectively.
argument
and ^ >
Lemma
10 in
the government
if
many
more short-sighted than
of important cases, political
will lead to additional distortions that will
We now
where politicians
in practice
realistic political
citizens, this part of
economy considerations
not disappear even asymptotically.
to support this
theorem (while the
full
proof
is
in
be the Lagrange multipliers on the constraints (28) and (29)
Appendix B shows that
V {w)
is
differentiable.
Furthermore, in
the text, we simplify the discussion by assuming that (C, L) slntA and w' eIntW. Therefore,
taking the
first
order condition with respect to w' and using the Envelope theorem,
we obtain
that
^V'{w')
The
other
=
-il>-j
v'{x){^P
that
(32)
V'{'w)-i;
order conditions with respect to C,
first
Uc + UL =
Equation
=
makes
we must have
i/)
>
it
'^v'iL),
+ 'r) =
L and x
(31)
imply:
and
Uc.
clear that aggregate distortions are related to
at
t
=
0,
(32)
(33)
ip.
otherwise the government would receive
27
evident
It is also
zi^o
=
initially;
which together with the sustainabihty constraint
cannot be a solution when
this
ip
=
0.
(29)
=
would imply Ct
Lt
—
for all
Uc + Ul >
Equation (32) then yields that
t,
and
0,
and
Proposition 2 implies that the marginal tax rate on the highest type will be strictly positive,
TN >
i.e.,
at
i
=
0.
Part 2 of Theorem 3 states that, as long as
<
/3
eventually aggregate distortions will
S,
disappear and marginal labor income taxes on the highest type
ways, this
see why, let us start with the case where
inequality above
This,
is strict
when
combined with the concavity
—
The
utilities for
why
intuition for
The
follows.
of the value function
V {•),
which
is
w
and w'
=w
ii ip
—
0.
Lemma
8 in
This shows that
Theorem
is
3.
as
incentives for the government in the current period are provided by both con-
promised
utility w. Therefore, future
and by consumption
incentives for government are backloaded.
similar to the reasons
(see, for
why
in future periods represented
by the
government consumption not only relaxes the sustain-
abihty constraint in the future, but also in
be useful
>
in
the rewards to the government are increasing (nondecreasing)
in the current period, x,
is
if ip
proved
the government are nondecreasing as stated in part 2 of
sumption
scheme
which case equation (31) implies
the sustainabihty constraint on the government (29) binds.
Appendix B, implies that w' > w, with w' >
the promised
in
6,
To
intuition clear.
= V'{w)-^<V'iw).
V'iw')
The
(3
many
tend to zero. In
makes the
a surprising result, but the structure of the model
is
will
all
The
prior periods.
Thus,
all else
intuition for this backloaded
in principal-agent
equal, optimal
compensation
models backloading compensation may
example, Ray, 2002).
Since promised values to the government form a nondecreasing sequence [wt (/i')}°^q and
compact
are in a
set
W=
mideately implies that
i/^
[0,
=
0.
w], they will converge to
some value w*.
If
w* < w,
(31) im-
In other words, the Lagrange multiplier on the sustainabihty
constraint of the government, (29), eventually reaches zero, and at this point, aggregate distortions disappear. Proposition 2 then implies that, as long as (C, L) eIntA, the marginal tax
rate on the labor supply of the highest type, 9;^, also vanishes as in the standard Mirrlees
program.
The
is
intuition for
why
the multiplier on the sustainabihty constraint eventually reaches zero
related to the fact that promised utilities to the government are increasing. Loosely speaking,
we can remove
sequence of
the sustainabihty constraints in the very far future, without influencing the
utilities
promised to the government
at
time
t
—
0.
This implies that eventually
the multipliers on these sustainabihty constraints must tend to zero.
28
Finally, let us consider the case with 5
<
agents, backloading incentives for government
which constraint
a steady state (C*, L*, x*)
is
l
C* + x*
with the steady-state
shows that when
6
<
Since government
becomes costly
utility of the
(3,
ever reached,
Theorem
is
>
it
^L*
&ndv{x*)
=
is
The
<
for
w,
slack. In
system of equations
(34)
.
{l-5)v{L*),
government equal to w*
it
w
Consider any
V'{w), and thus w'
will solve the following
—
v{L*). Equation (34) immediately
would feature a positive labor
distortion,
intuition for the presence of (asymptotic) aggregate
directly related to the fact that
when
the government
than the agents, backloading does not work. Since backloading was
on the
patient than the
^=fl-^)^^^
+
3.
less
the sustainability constraint, (29),
were a steady state reached,
as claimed in part 3 of
distortions in this case
when
is
for agents.
does not bind. Then (31) implies that V'iw')
(29)
so that promised utilities will be decreasing
fact, if
p.
patient
is less
essential for the multiplier
sustainability -constraint (29) going to zero, this multiplier remains positive,
and the
additional distortions created by the sustainability constraint remain even asymptotically.
An Example
5.2
We now
for
an Economy with Private Histories and
illustrate the results
i.e.,
u{c,l\e)
m
is
—
{^Oi^i} and
= ^c--^,
(35)
a parameter determining the relative disutility of labor.
that type ^o
is
=
disabled and cannot supply any labor, so ^o
Let us also assume that a fraction n
function of the government
is
Capital
from the previous subsection with a simple numerical example.
Consider an economy with two types,
where
No
=
1/2 of the population
also given
by v
Since type ^o cannot supply any labor,
compatibility constraint for type 6i
(x)
=
is
0,
We
continue to assume
and we normalize
of type 9i
and that the
we have
/
{9i)
=
L/ir.
computed
directly
(1
—
1.
utility
Moreover, the incentive
is
(36)
c{9o) and c{9i) can be determined as solutions to (36) holding as equality
the resource constraint,
=
y/x.
V^T^-^>^^I^•
Then
9i
it)c{9o)
+
ttc{9i)
and used with the program
=
C.
Given
this structure,
U{C,L) can be
(27) to derive the value function
29
and to
V {w).
Figure
We
and
1:
Theorem
3 part
2.
Value function with
take a baseline case where the government
m=
In Figure
5.
The
U-shaped.
1
we show the shape
incrccising part
is
is
j3
fact that
if
level of utility, the sustainability constraint will force the
is
everywhere decreasing. In
mechanism, the economy
will start at the
fact, as
=
0.9.
Note that
the government
economy
is
P
—
V (w)
5
is
—
0.9
inverse
given too low a
to produce a very low level
V
of output. For this reason, the relevant part of the value function
the peak, which
5
as patient as the citizens,
of the value function.'^^
due to the
=
{w)
is
the segment after
noted above, with the best sustainable
peak of the function
V (w).
Next, we show the time path of normalized promised values to the government (defined as
(1
- S)w) and aggregate
distortions both for the baseline case, 6
of lower discount factors, S
=
0.8, 0.7,
and
2,
when
in these
respectively, for 5
5
=
3,
Proposition
case where
^'^We
{wt}
is
0.7, 0.8
0.9.
is
2, is
Consistent with the results in
everj'where increasing even
also equivalent to the marginal tax
S,
and consistent with part
interesting feature of the
2 of
example
is
and
also for a range
lowest curves
on type
Theorem
3,
^i).
The
is
5
<
for S
Theorem
level w*.
when
in turn, depicts the evolution of the aggregate distortion, 1
P —
An
and
The
an increasing sequence and converges to some
examples the sequence {wt}
Figure
to zero.
jS,
=
0.9,
Figure 2 plots the time path of the promised
0.6.
value to the government, w, for these four different cases.
and then,
—
=
0.6,
3 part
Interestingly,
p.
+ Ul/Uc
(which, from
lowest curve shows the
the aggregate distortion converges
that the convergence of {wt} and of distortions
compute everything non-recursively, assuming that a steady state is reached after T periods and then
more accurate than the recursive approach in tlie
solving a sequence problem, which turns out to be faster and
computations.
30
r
10
Figure
/3
=
0.9
2:
Time path
and b
—
15
20
25
30
of normalized promised values to the government, {(1
0.9; 0-8; 0.7; 0.6.
Higher curves correspond to higher values of
—S)wi^, with
<5.
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
-0.05
Figure
3:
Time path
of distortions with
correspond to lower values of
/3
—
5.
31
0.9
and b
=
0.6, 0.7, 0.8, 0.9.
Higher curves
18
16
14
12
I 10
o
a
6
4
r
r
2
Figure
4:
Time path
to lower values of
is
of xtjYi with
/3
=
0.9
and 8
—
0.6,0.7,0.8,0.9. Higher curves correspond
b.
rather fast. This suggests that the best sustainable mechanism
may converge
to a
mechanism
without aggregate distortions and with zero marginal tax rate on the highest type agents very
rapidly.
The
with
/3
figure also
=
(5,
when
shows that, as predicted by part 3 of Theorem 3 and
< P
6
in contrast to the case
aggregate distortions do not disappear asymptotically; in fact, they
could be quite sizable. For example,
when
=
6
0.6,
the aggregate distortion converges to an
asymptotic value of 0.15 (the highest curve in the graph).
A
final interesting feature is
how much
government. Figure 4 shows this again
of the
government
is
for 5
of the
—
economy's output
0.9, 0.8, 0.7,
and
0.6.
is
being captured by the
When
the discount factor
equal to that of the citizens, even in the asymptotic equilibrium, the
government receives a very small fraction of the output (and w*
previous subsection). As
we consider lower discount
<w
with the notation in the
factors for the government, its temptation
to deviate increases, so a higher fraction of the output goes to the government, but even with
b
—
6
6.1
0.6, this is
only
16%
of total output.
Optimal History-Dependent Sustainable Mechanisms
Characterization of Best Sustainable Mechanism
We now
return to the analysis of the general problem in Sections 2-4 without the restriction to
private histories, and
we
also re-incorporate capital. Despite the fact that individual allocations
32
now be a function
will
of the entire history of reports
by the individual, the analysis
will parallel
the discussion of the best sustainable mechanism with private histories in Section 5 (and from
Theorem
The
1,
individuals will continue to report their types truthfully).
program was defined above
quasi- Mirrlees
and Theorem
in (14),
In addition, Appendix
the best sustainable mechanism solves a quasi-Mirrlees program.
shows that ZY({Ct,-Lf}^o)
that
some
we can think
of variations in sequences {Ct,Lt}'^Q
We
specific s is varied.
where only one element, Cg or L^
U
denote the derivative of
derivatives of the production function with respect to labor
We
for
with respect to such variations by
Wc^({Ct,I/t}^Q) and WL^({Cj,Lt}go) °^ simply by Uc^ and Ul^.
and Fk^-
C
{Ct,Lt}go G A°°. This implies
differentiable in the sequences
i^
2 established that
We
also denote the partial
and capital
by Fl^
at time s
next extend our definition of an undistorted equilibrium to this more general
environment:
Definition 5
We
say that the (potentially stochastic) sequence {Ct
induced by the best sustainable mechanism T*
is
= Cf
say that [Ct
(ft*'),
(ft*)
it
,
(ft*)
surely) undistorted as
As
U
t
,
(ft*')
= Lf
-^t+i
(ft*)
undistorted at
[Xt (ft*)}^o
a solution to (17) subject to (18) with
Ct> (ft*')
is
(ft*'), lit'+i (ft*')
,xt (ft*)}^o
*''
~
=
t'
(/i*)
if <
Ct
,
Lt
(ft*)
(ft.*)
Lt
,
^°"''
{^* (^Olt^o
,
(ft*)
""^^
i^t'+i (ft*') for all
asymptotically undistorted
i^t+i
^'
ft*'
(ft*)
Kt+\
,
^ ^*
e
i7*'.
,
Xt
(ft*) \
'^'^^
We
(almost
if it is
^ oo.
the natural generalization of Definition 4 to this more general environment.
This definition
is
in Definition 4,
when
{Ct,
Lt}^Q 6lntA°°, the current
definition implies that
an undistorted
allocation will satisfy
i^Ct
at time
time
t
t
(or as
^
t
Fu =
-^Lt and Fk^^^
Here, the
oo).
first
Uc,+^
=
Uc,
(37)
condition requires the marginal cost of effort at
given the utility function U{{Ct, Ltjj^o) ^o ^^ equal to the increase in output from the
additional effort times the marginal utility of additional consumption, and
generalization of condition (26) from the
economy without
capital.
is
thus an immediate
The second one
requires
the cost of a decline in the utility by saving one more unit to be equal to the increase in output
in the
Once
next period times the marginal utility of consumption then.
again, these are
aggregate conditions since they are defined in terms of the utility functional
which represents the ex ante maximal
Moreover,
if
utility of
clear that {Ct
(ft*)
,
Lt
(ft*)
,
Kt+i
(ft*)
,
Xt
(ft*)
Z//({C(, LjI^^q),
an individual subject to incentive constraints.
a steady state exists and the conditions in (37) hold as
}J^„
i
must be undistorted.
—
*
oo,
then
We therefore
it
is
also
say that
there are no asymptotic aggregate distortions on capital accumulation (or no aggregate capital
33
(ft*)
} j^q
.
taxation)
.
if -Fa',+i -l^Ct+i
— ^Ct
and no aggregate distortions on labor supply ifUct '^U
Let us use the notation {C,L} 6 A°°
Lt
ih'')
L
-^
if
{Q
(h^) ,Lt
almost surely, and the steady-state pair {C, L)
[Ct {h'),Lt {h'),Kt
(/i*)}
[C\L\K*)
-^
w =
assumption
is
inaxf(^£,ig^oo_/(->o v
is
{F [K, L) —
= Wa({a
C — K) / {1 —
L)-C -K)/{1-
Then
sustainability) There exists [C,L,K)
such that v [F {K,L)
5)
the main result of this section parallels
Theorem
and
L
Theorem
3,
/ {1
but
is
When
sustainability
when
satisfied:
is
e argmaxr^^^ig^^oo
-
>
S)
(F
v
weaker
in
(/?,
some
/^'>q
L))
respects:
4 Consider the model with the general environment and suppose that Assumptions
1-5 and 7 hold and that there exists {Ct
t
- C - K)
and
level of utility
The key
5).
C
,L, (/i*)}£o)
a generalization of Assumption 6 to this environment and requires that
Assumption 7 (general
{F {K,
(/i*)
maximal steady-state
the government receives a utility of w, the sustainability constraint, (12),
V
-^
asymptotically feasible. "^^
almost surely, define W^^
Finally, with analogy to the previous section, define the
for the ruler as:
Q [h^]
{h^)}^o ^ ^°° and
= —^U
all h^
6
some
H*- for
There exists
t'
labor supply at
<
(/i*)
Lt (/i')}^o ^IntA°° (with Lt
,
positive probability
(/i*)
>
for some
H^ C H* ).
oo sueh that there are aggregate distortion on capital accumulation and
t'
Let r* be the best sustainable mechanism, inducing a sequence of consumption, labor supply and
capital levels {Ct
[Ct
(/i*)
,Lt
(/i*)
(/i*)
,Lt
,Kt+i
(/i*)
(/i')}
p\\mt-,co Q~*^Uc ~'^)' where
^-
If
f
'^ ^-
,Kt+i
—
>
(/i*)}-
Suppose a steady state
{C*,L*,K*) almost
^<
1.
surely.
exists
Moreover,
such that as
let if
=sup{
t
—> oo,
g G [0,1]
:
Then:
then (almost surely) 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.
we no longer have the equivalent of Proposition 2, since without specifying the stochastic
we cannot estabhsh the equivalent of the single-crossing property in consumption sequences.
This problem is common in models of dynamic taxation, even with the full commitment assumption, see, for
example, Golosov, Kocherlakota and Tsyvinski (2003).
^^Notice that
process for
di^t,
^^Note that despite the similarity of the symbols, A°° and A°° are very different
vector space of bounded infinite sequences, ^oo, while A°°
34
C
R'^.
sets.
A°°
is
a subset of the
Proof. See Appendix D.
The major
is
results
from Theorem
that instead of comparing the government discount factor,
5,
rate at which the ex ante marginal utility of consumption Uq^
denoted by f. Clearly, in the case where W({Ct,I/t}^o)
the rate at which
Uq
declines in steady state
theorem are closely related to those
The most important
3 continue to hold here.^°
in
discount factor" of the citizens, since
it
is
Theorem
to
we now compare
it
to the
declining in the steady state,
is
^^™® separable as in Theorem
'^
exactly equal to
3.
/3,
difference
Moreover,
/3,
so that the results in this
in reality,
if is
the "fundamental
measures how one unit of resources at time
with one unit of resources at time t+1 (from the viewpoint of i
=
Only
0).
3,
t
compares
in special cases (e.g.,
without any dynamic incentive linkages) does this fundamental discount factor coincide with
p. Therefore, the case of
is
first
U{{Ct,Lt}^Q),
be present at
citizens.
a distortion, though because of the ex ante utility function of the
t
=
0.
is
nonseparable,
we can no longer be
The most important
sure that this distortion will
results are again contained in parts 2
the theorem. Part 2 states that as long as a steady state exists and
rapidly, the multiplier of the sustainabihty constraint goes to zero.
the sequence {Ct, Lt, Kt}'^Q
and capital accumulation
capital
and private
hand, states that
if
is
greater than in
distortions. This generahzes the results
histories to the
much more
since
3,
we provide a
Since the objective function
mechanism
in this case,
we
is
This establishes that
it
from the economy with no
The
is
sufficiently low,
3,
on the other
then aggregate
significance of this result
is
even
implies not only additional distortions on labor, but also
positive aggregate capital taxes in contrast to the existing literature
again,
3 of
dechnes sufficiently
general environment here. Part
the discount factor of the government S
Theorem
Uq
and
asymptotically undistorted, with no aggregate labor supply
distortions will not disappear, even asymptotically.
Once
which the government
part of the theorem states that the sustainabihty constraint of the government,
(12), necessarily introduces
citizens,
5 indeed corresponds to the situation in
more patient than the
as patient as or
The
<
ip
heuristic
argument
on dynamic
fiscal policy.
here, leaving the proof to the
Appendix.
no longer time separable, to characterize the best sustainable
follow
Marcet and Marimon (1998) and form a Lagrangian of the
form (again suppressing dependence on history
/i*
for notational simplicity):
oo
max
{C(,Lt,/f,,xt}(^o
C = U{{CuLt}^o) + ^S'{f,A^t)-{f,t-Ht_MF{Kt,Lt))}
(38)
^^^
^°The parts that are missing from this theorem relative to Theorem 3 are that the aggregate distortion is
(instead it could be at some later date), that the sequence of promised values to the government is
t =
increasing and a statement that a steady state exists. Moreover, as noted in footnote 38, there is no equivalent
of Proposition 2 in this case. Finally, the theorem imposes the rather weak assumption that there exists
for some t and some h\
{Ct (/i') Li (/i')}^o eIntA~ with Lt {h') >
at
,
35
subject to
Ct
for all
where
t,
constraint
The
|J,^
—
+
Hi^i
with
Kt+i
ix_i
—
< FiKt,
and
5^'^p^
Lt),
and
>
is
(39)
the Lagrange multiplier on the
(12)."*^
differentiability of
Ul,
Ua =
Since
V'f
+ xt +
/Xj
>
fj^t^i,
^({Ct, Lij^o) implies that
-
-
ii,.,)v'{F{Kt, L,))Fl,
-
<5^(/it+i
5*(m,
[Wc,+,
for
-
{CuLt]'^^ GlntA°°, we have:42
^
-He,
Fl,
(40)
^^t)v'iF{Kt+uLt+l))] Fk,^,
(41)
we obtain that
-UL,<UcrFu,
and
(42)
lla<Uc,,^,-FK,^,.
(43)
These two conditions imply that there may be positive distortions
accumulation
be too high
The
—
if
the inequalities are
(relative to the full
strict,
^'-^''-''
construction,
go to
infinity.
/Ltj
is
an increasing sequence, so
Suppose that
[Ct, Lt,
allocation).
on the other hand,
xt,
6'v'ixt)
By
supply and capital
the marginal product of labor and capital would
commitment Mirrlees
with respect to
first-order condition
in labor
it
yields:
5'^\'{x,^,y
must
either converge to
Kt) converges to some {C*,L*,K*)
some value ^*
—and
or
xt converges to
^'To derive (38), form the Lagrangian
oo
f2S''v{x,+,)-v{FiK„L,))
(=0
then note that
''
t=0
s=0
t=0
Hi^i + ipt with fj._i = 0. Substituting this in £.' above gives (38).
To obtain these equations, let the multiplier on constraint (39) at time i be
where
/i,
=
with respect to Ct gives Uci
=
Substituting for Kt gives (40).
-
The
first-order condition with respect to
S' (m,
-
=
Wt,
-5'(Mi+i
Kt.
Then
the first-order condition
while the first-order condition with respect to Lt gives
Kj,
M,_,) v'{F(J<t,Lt))FL,
-k,Fl,.
Kt+i, on the other hand, gives
- IJiW{F(Kt+uLt+i))FK, + + Kt~ Kt+iFK,^,
,
Substituting for Kt and Kt+i and rearranging gives (41).
36
=0
'
=
X*
L*
— C* — K*
Theorem
the proof of
is
Uq
proportional to some
is
some value
surely) converges to
Uq
If
.
<
ji*
oo,
if
and that both
<
then we can show that ^^ (almost
S,
and
(42)
(43)
must hold as equality
(see
establishing the result stated in part 2 of the theorem. In contrast,
4),
proportional to some
>
ip
then
5,
/if
if
tends to infinity and aggregate distortions do not
disappear.
Example
6.2
We now
History-Dependent Mechanisms
for
Theorem 4 and show how
briefly illustrate the results of
defined as sup{p e
the agents,
[0, 1]
plimt_oo Q~^Uq
:
In particular,
/3.
—
0}
is
economy without
"almost constant types" as explained below. There are two types
function
cases,
ip
again equivalent to the discount factor of
us consider the following
let
some simple
in
—
capital
{6'o,^i}
and with
and the
utility
is
u{c,i\e)^u{c)-x{i/d),
where u
continuously differentiable, increasing and strictly concave and
is
differentiable, increasing
and
strictly convex.
Furthermore, suppose that u
conditions, so that first-order conditions are always satisfied as equality.
the low type
an individual
With
is
is
x
is
continuously
satisfies
Inada-type
We take 9q =
0,
so that
again disabled and cannot supply any labor. Suppose that with probability
born as high type, and remains so with
probability
1
—
individual
tt,
almost constant types, we
mean
is
(iid)
probability
1
—e
in every period.
born as low type, and remains low type
the limit of this
economy
as £
^
0."^^
Then
tt
forever.
By
the quasi-Mirrlees
formulation can be written as
oo
U{{CuLt}Z^)=
i^ax
oo
^^/3Mn(ct(0i))-x(^t(0i)M)]+(l-^)E/3M^(ct(^o))],
(45)
subject to
TTCt (^i)
+
(1
—
tt) ct
{9o)
<
nit (6'i)
—
Xt for all
oo
oo
5^/3*
[t.
(ct (0i))
-
X-
(U^i)
M)] >
t=0
where L(
i,
=
nit (^i)
and
t,
E^* [^
(^t (^o))]
t=0
and Ct
=
Lt
while the second constraint
is
—
Xt-
The
first
constraint
is
the resource constraint for each
the incentive compatibility constraint sufficient for the high
type to reveal his identity given the presence of effective commitment along the equilibrium
path. Because e
''
If
instead
—
>
0,
we use the
we do not
alternative
specify other incentive compatibility constraints. Assigning
game form
outlined in
types rather than almost constant types.
37
Remark
1,
this
example would work with constant
0.35
'
0.3
•
0.25
0.2
-
0.15
-
0.1
-
0.06
-0.05
Figure
5:
-
The time path
0.6,0.7,0.8,0.9. Higher curves correspond to lower values of
Lagrange multipliers A and
this
=
of distortions for ahnost constant types with
=
and 5
S.
to these constraints, the first-order necessary conditions of
/3*/Ut
problem can be written
0.9
as:
+
(7r
-
A)u'(ct(0i))
= (1-
n-X)u'{ct{eo))
(46)
^^^lt
(47)
l/^t,
and
^-^x'{k{Or)/e,]
Mf
(48)
Equations (46)- (47) imply that
u'ict{0i))
u'icti9o))
Consequently, there
is
It
{Oi) -^
in exactly the
r, and hence
same form
the derivative Uq^ declines
factor of the citizens,
i.e.,
—
>
/j.^
as
is
(/?
x*,
—*
(46)-(48)
fi*.
''''in
=
13,
+
X)
combined imply that
Consequently, in this case
Theorem
3.
=
/?.
It is
tp
all
periods. Moreover,
q
(^i)
—
j3,
—
so
>
c^*,
q
if
a
(Oq) —> c^*,
Theorem 4
applies
Therefore, in this particular case, the rate at which
easy to determine, and
generalizes to the case where there are
ip
(tt
constant risk-sharing between the two types in
steady state exists, so that Xt
and
A)
fl
it
does so at the same rate as the discount
also straightforward to see that the
more than two
same argument
types. "^^
we conjecture that whenever there exists a stationary distribution of consumption among individuals,
though we have not been able to prove tliis conjecture.
fact,
38
For a numerical illustration, we again consider the
same parameter values from subsection
the
distortion,
1
-\-UijUci-, in Figure
For brevity, we simply show the aggregate
5.2.
Consistent with part 2 of
5.
with two types and
utility function (35)
Theorem
when
4,
j3
—
5
=
0.9,
the lowest curve shows that the aggregate distortion converges to zero and the convergence
again rather
fast.
when
Instead,
<
5
is
the aggregate distortion converges to a positive, and
/3,
potentially large, asymptotic value.^^
7
Benevolent Time-Inconsistent Governments
The
analysis so far
Although
was
simplified
by the
this case is of relevance for
to understand
how
fact that the
many
political
government was purely
economy
applications,
it is
self-iiiterested.
also
important
the results generalize to the case considered by Roberts (1984), Freixas,
Guesnerie and Tirole (1985), or Bisin and Rampini (2005), where the government
benevolent, but "time inconsistent",
do
this,
unable to commit to a
i.e.,
we now consider a more general
is
function for the government of the form:
utility
and a >
function
is
(the case with a
=
t
to the fully-benevolent case
we need
standard assumption
in
when
5
~
+
s,
t
(49)
+
s (with
which includes the public history
identical to our analysis above). Therefore, this utility
is
identical to that of a purely-self-interested
In this case,
{9'+')]],
the average (expected) utility of the citizens at time
expectations based on information available at time
/i*"*"*),
still
dynamic mechanism. To
full
E(^(5^ {l-a)v{xt+s) + aUt+s I u[ct+s,k+s\0'+') dG'+'
where the second term
is
f3
and a
=
government when a
=
0,
and
identical
1.
to strengthen
Assumption
most analyses
of
and assume separable
1
dynamic taxation
(e.g.,
utility,
which
is
a
Golosov, Kocherlakota and
Tsyvinski, 2003, Kocherlakota, 2005).
Assumption
1'
=
(separable utility) u{c,l\6)
w
(c)
—x
(^
|
continuously differentiable, strictly increasing and concave, and xi'
ferentiable, strictly increasing
9
and convex for
all
6
& Q, and
where u
^)>
I
^)
satisfies
x
^s
(0
:
M+
—*
R
continuously
|
0)
=
for
is
difall
ee.
The next theorem shows
that
Theorem
general environment, but with Assumption
^
5.2.
Note that the distortions are the same here as
The reason
When
1
1'
and Proposition
1
continue to hold in this more
replacing Assumption
for the case
1:
with the private histories studied in subsection
that without the sustainabiUty constraints, the problems would have the same solution.
sustainability constraints are present but satisfied, there are no other reasons for the solutions to difli'er.
is
39
.
Theorem
4
and 5
5 Suppose that government
utility is
Then for any combination of
hold.
mechanism, there
exists
given by (49) and that Assumptions
strategy profiles
V and a
1
2,
',
3,
that support a sustainable
another pair of equilibrium strategy profiles T* and a*
—
{a*
a')
\
for some a' such that T* induces direct submechanisms, a* induces truth telling along the
=
equilibrium path, and c[T,a,h]
c\T* ,a*,h], l[T,a,h] —l[T*,a*,h] andx[T,a,h]
Moreover, the best sustainable mechanism
is
=
2;
[F*
,
a*
,
/i]
a solution to maximizing (9) subject to (10), (11)
and the government sustainability constraint:
^tY.5'[{l-a)v{xt+s{h'+')) +
s=0
j [u (c (0*+^ h'+^)) - X
a (e,+,
{I
(0*+^ /i*+0
(^t+s)] dG'-^^
I
max
{l~a)v[xt) + a [ uict(e\h^))dGHe^),
all t
and
fall h*
Q
(50)
J
x[+fc[{B\h*)dG*{e')<F{Kt(h*^-^),Lt(h^))
for
{e'+^i
H''.
Proof. See Appendix E.
The
idea of this theorem
is
that the
same type
of
punishment strategies that were used
the case of the purely self-interested government also work
when
the government
is
In particular imagine that the government has undertaken a deviation in which
some
of
clearly
its
in
benevolent.
it
has used
past information in order to improve the ex post allocation of resources. This could
be desirable given the
utility function of the
the Roberts' (1984) example,
best sustainable
mechanism
same punishment
it
may have
will
To
establish
in (49),
but as illustrated with
very negative consequences ex ante. Therefore, the
have to discourage such deviations. To do
strategies as above, in
supply zero labor.
government
which following any type of deviation,
Theorem
strategies are sequentially rational.
this,
When
5,
all
all
we need
to
show
is
all
imagine the
individuals
that such punishment
other agents choose zero labor supply, following
any deviation to positive labor supply, the government would consume some of the increase
in output
itself,
utility function
and would redistribute the
assumed
in
Assumption
1'.
rest equally
Since there
this implies the deviating individual will receive
positive labor,
and thus
it is
among
is
all
agents given the separable
a very large number of citizens,
no additional consumption from supplying
sequentially rational for
all
citizens to supply zero labor following
a deviation by the government.
This theorem therefore shows that revelation principle appHes to the case of benevolent, but
time-inconsistent governments as well, though under the additional assumption of Assumption
r.
The next example shows why
this
assumption
40
is
necessary:
Example
To avoid
1
economy with n agents
be constructed
6
€
where with Xi
in labor
=
=
utility u{c,- \9
(')
Now
u
strictly increasing in
who can
is
—
1 is
u
only supply
{c,l
=
6
\
=
1)
=
I
and has
0,
u{c — Xi
Furthermore, suppose that aggregate output
/.
fully benevolent,
i.e.,
a
=
1
in
terms of the
is
(0)i
Hnear
utility function
imagine the economy has entered the punishment phase where each citizen
I
=
and consume
<
(^')
consumption (output
distribute
same example can
large (exactly the
is
while the utility of type 6
(c),
V > Q such that Xi
to
1
=
Gi)
supposed to supply
=
n
Q corresponding to the disabled type,
and that the government
in (49).
9
example, where
for this
of agents, let us consider a finite
an economy with a continuum of agents). There are two types of agents,
in
with 6
{0, 1},
among continuum
issues of deviation
I'
>
=
c
0.
Consider a deviation by an agent,
i',
is
of type
Following this deviation, the benevolent planner will
1-
maximize
0) to
its
own
utility,
which involves maximizing
average utility of the citizens, thus equating the marginal utility of consumption across agents,
i.e.,
u' ic)
thus,
Ci
=
Ci'
-
{I'
—
Ci -^
Xi
{!-'))
Xi
{!')
/" and
for all
Cj'
=
u' {a-
¥=
i
{I'
=
-
Xi
-
Xi
The
^'-
{I'))
/n
{I'))
ioralli^i'
resource constraint
+ Xi
('')
The
is
(n
—
1)
q +
Q'
=
/',
resulting utihty of individual
or
i' is
ui{l'-xiil'))/n)>u{0),
any
for
n, thus giving
him
greater utility than supplying zero labor.
punishment phase where each
rational
citizen
is
This proves that the
supposed to supply zero labor
and thus cannot be part of a (Perfect Bayesian) equilibrium with
The next theorem
provides a generalization of
Theorem
4 for the
is
not sequentially
this utility function.
most-commonly studied
case where types are constant (in our context, types are "almost constant" as in subsection
and
6.2)
(3
Theorem
tions 1
',
and au'
=
6 Suppose that government
2, 3,
(0)
S.'^^
^
4
and 5
{1
—
hold.
utility is
given by
(4-9)
with a 6
(0, 1)
and
that
Assump-
Furthermore, assume that there are (almost) constant types,
a)u'(O).
(3
—
6
Then, asymptotically there are no aggregate distortions on labor
supply and capital accumulation.
Proof. See Appendix E.
This theorem implies that in an economy with (almost) constant types, aggregate distortions disappear irrespective of the degree of benevolence of the government.
Once again with the game form
in
Remark
1,
this
theorem can be stated
41
Consequently,
for constant types.
there will be no aggregate capital taxes and no further taxes on labor beyond those implied
by the full-commitment Mirrlees
arbitrarily close to the fully-benevolent case,
Roberts (1984), where
distortions.
Once
in
main source
again, the
it
have so
economy
of the difference
in
is
the infinite-horizon nature of our
which the government
be punished
will
if
Although
versus Mechanisms
behavior of the best sustainable mechanism under political
far characterized the
constraints.
results in
gathers via the earlier submechanisms.
Anonymous Markets
We
is
a very similar environment, the equilibrium always involved extreme
exploits the information
8
the government
1,
>
and the theorem contrasts with the
economy, which allows us to construct equilibria
it
—
In the case where a
economy.'*''
this
was
largely motivated
by our objective of understanding
the form of optimal policy in an environment with both informational problems on the side
of agents and selfish behavior on the side of the government, an additional motivation
investigate
when
in
anonymous markets.
restrictions preclude a detailed discussion of
we take the
simplest conception of
In this section,
we begin
how anonymous markets
anonymous markets
as one in
this analysis.
we do not need
to
assume anything
specific
Space
should be modeled, so
which there
is
no intervention
by the government, and consequently more limited insurance. For the purposes
in this section,
to
mechanisms and when
certain activities should be regulated by (sustainable)
they should be organized
is
of the exercise
about how the anonymous markets
work, except that there exists a well-defined anonymous market equilibrium, which yields ex
ante utility
note
is
TJ''^^^
that U'*^*^
to individuals before they
is
The
point to
imposed on the government
(since there
is
no government involvement
anonymous markets).
Given
this,
we can provide some simple comparisons between anonymous markets versus
sustainable mechanisms. Throughout this section,
allocations on public histories to simplify notation.
we suppress dependence
Our
first
that an increase in the discount factor of the government,
tive relative to markets.
Let U"^^^
(S)
Proposition 3 Suppose V^'^'
6^0, V^^
(5)
>
5,
of strategies
and
comparative static result states
makes mechanisms more
attrac-
be the ex ante expected value of the best sustainable
mechanism when the government discount
as
their types.
independent of both the discount factor of the government and any other
institutional controls
in the
know anything about
factor
is
5
and U'^^ be as defined above.
U-^^'^ then V^^'^ {5')
>
U''^-'^^
for
all 6'
>
5.
Moreover,
>U^^^((5).
''^The assumption that au' (0)
^
(1
-
a) v' (0) rules out a special case in
work (though other more complicated approaches may work even without
42
which our method of proof does not
this assumption).
.
Proof. Let
is
be the feasible
set of allocation rules
when
the government discount factor
equal to 6 (meaning that they are feasible and also satisfy the sustainability constraint (12)).
Let
It
S (5)
{ct {5)
,
{5)
It
xt
,
(5)}^o ^
and labor supply across
are vectors consumption
(6)
have
{ct {5)
hand
side of (12)
,
k
(S)
,
xt {5)}'^q
is
S
£
(5')
,
(5)
It
xt ((5)}^o
,
the government's discount factor
U^^,
therefore
U^^^
The second part
{5')
> U^^
(S)
is 5'.
and
/°
Since
>
6'
is
unchanged, so {q
is
and
feasible
ct {6)
we immediately
5,
(5)
It
,
the
is 5'
,
(5)
xt ((5)}^q
,
yields expected utility
This implies that U'^^^
and
left-
U'^^
{6)
large as
(5') is at least as
> U-^^.
follows from the observation that with
always achieve the autarchy allocation, thus
c"
types.
—when the government's discount factor
higher, while the right-hand side
satisfies (12). Therefore, {ct (5)
when
represent the best sustainable mechanism, where
"^ (^)
U'^^ > Eq
anonymous markets,
[X^t^o /^*^ i^° i^t)
denote the optimal autarchy choices of an agent with type
the centralized mechanism leads to utihty of Eq
[E~o I^^u (0,
|
1°'
>
individuals can
[St)
\
^t)]
where
In contrast, with 5
9.
< Eq [EZo Z^*'" (c'*
Ot)]
,
—
{Ot)
>
,
0,
/" {Ot)
m
An important
implication of this proposition
and unable to commit to policy sequences, not
intervention can be achieved by a
shown by the
last
is
all
that because the government
is
self-interested
equilibrium allocations without government
mechanism operated by the government. Consequently,
part of Proposition
3,
anonymous markets can be
as
preferred to sustainable
mechanisms.
Let us next consider a modification of our main setup along the
whereby the government can consume only a portion
17,
77
lines
mentioned in footnote
of the output
77,
corresponds to better insututional constraints on government behavior.
of the
mechanism
government.
We
as
U
(?]),
now
as a function of the institutional restriction
on the
(77)
> U-^^,
then V^^^^
(77')
> U-^^
for
all r^
<t]. Moreover,
U^^(r7)>U^^^
The proof
of this proposition
is
similar to that of Proposition 3
the intuitive result that better institutional controls on government
desirable relative to markets.
ized
Define the value
then have:
Proposition 4 Suppose V^^^
asrj-^0,
i.e.,
so that a lower
It also
and
is
omitted.
It states
make mechanisms more
implies that with sufficiently good controls on central-
mechanisms (government behavior), sustainable mechanisms are preferred to anonymous
markets.
43
\
Ot)]
Concluding Remarks
9
The optimal
taxation literature pioneered by Mirrlees (1971) has generated a
number
of im-
portant insights about the optimal tax policy in the presence of insurance-incentive trade-offs.
The
dynamic taxation
recent optimal
literature has extended these insights to a macroeco-
dynamic behavior
nomic setting where
issues of
criticism against
all
of this literature
government with
full
A
of central importance.
A
potential
relevant and important question in this context
whether the insights of this literature apply to
own
is
that these optimal tax schemes assume a benevolent
is
commitment power.
reelection, self-enrichment or their
of taxes
real world situations
where
politicians care
commit
individual biases, and cannot
is
about
to sequences of
future policies or to mechanisms.
This paper investigated this question and characterized the conditions under which these
insights hold even
when mechanisms
are operated by self-interested politicians,
the resources and the information they collect.
tion
by the government
potential misuse of resources and informa-
(politicians or bureaucrats)
makes mechanisms
less desirable relative
mechanism design approach, and implies that
to markets than in the standard
cations resulting from
The
who can misuse
anonymous market transactions
will not
certain allo-
be achievable via centralized
mechanisms. Nevertheless, centralized mechanisms may be preferable to anonymous markets
because of the additional insurance they provide to risk-averse agents.
The main
contribution of the paper
is
an analysis of the form of mechanisms to insure
idiosyncratic (productivity) risks as in the classical Mirrlees setup in the presence of a
interested government.
Given the
infinite horizon
nature of the environment in question, we
can construct sustainable mechanisms where the government
An
resources and information.
self-
is
given incentives not to misuse
important result of our analysis
is
the revelation principle
along the equilibrium path, which shows that truth-telling mechanisms can be used despite the
commitment problems and
this tool,
The
the different interests of the government and the citizens.
we provide a characterization
of the best sustainable mechanism.
other major results of our analysis are as follows. First, under fairly general conditions,
the best sustainable mechanism
in
Using
which the ex ante
utility of
is
a solution to a quasi-Mirrlees problem, defined as a program
agents
is
maximized subject
constraints as well as two additional constraints
on the
to incentive compatibility, feasibility
total
amount
of
consumption and labor
supply in the economy. Second, under additional conditions, we can characterize the
and asymptotic
distortions created
the government
is
initial
by the best sustainable mechanism. In particular, when
sufficiently patient (in
many
cases as patient as, or
more patients than,
the citizens) we can show that the Lagrange multiplier on the sustainability constraint of the
,
44
government goes to zero and aggregate distortions disappear asymptotically. Consequently, in
the long run the highest type individuals will face zero marginal tax rate on their labor supply
as in classical Mirrlees setup
and there
dynamic taxation
These
from Mirrlees'
literature.
classical analysis
be no aggregate capital taxes as
will
latter results therefore
and from the dynamic taxation
asymptotically.
is
literature
may
follow despite
However, we also
not sufficiently patient, aggregate distortions remain, even
many
In this case, in contrast to
existing studies of optimal taxation, there
be positive distortions and positive aggregate capital taxes even
will
the classical
imply that some of the insights
the presence of political economy constraints and commitment problems.
show that when the government
in
in the long run.
In addition, assuming that individual preferences are separable between consumption and
leisure,
we
also generalized the results
on revelation principle and direct mechanisms to an
environment with potentially benevolent preferences
for the
government (including the
fully-
benevolent time-inconsistent case), and showed that our main characterization result applies
with (almost) constant types.
We
view this paper as a
first
step in investigating political
are both technical and substantial issues
left
economy
of mechanisms.
unanswered. First, we would
There
like to generalize
the results on time-inconsistent fully-benevolent government to non-separable utility functions
and to
richer
dynamics of individual types.
detailed comparison of centralized
ment misbehavior
to
more
Secondly,
it
is
important to undertake a more
mechanisms subject to commitment problems and govern-
reahstic models of
anonymous markets. Finally and perhaps most
importantly, our investigation highhghts a difficult but interesting question;
society be structured so that the
government (the mechanism designer)
how should
the
easier to control. In
is
other words, the recognition that governments need to be given the right incentives in designing
mechanisms opens the way
for the analysis of
"mechanism design squared" where the structure
,
of incentives and institutions for governments and individuals are simultaneously determined
(for
example, in the form of "constitutional design").
when we want
This becomes relevant in particular
to think about the interaction of different types of institutions in society, for
example the difference between contracting
institutions that regulate the relationship
between
individual citizens versus "property rights institutions" that regulate the relationship between
the state and individuals (Acemoglu and Johnson, 2005). While the existence of these distinct
types of institutions have been recognized,
not been investigated.
We
how they should be simultaneously designed has
believe that the approach and tools in this paper will be useful to
address this class of questions.
45
Appendices
Appendix A: Some Technical Results and Definitions
10
We now
We
G'
G
is
of technical results and definitions that will be used in the rest of the
take the definition of regular point from Luenberger (1969, p. 240).
X
Definition 6 Let
that
number
provide a
appendix.
Z
and
be
Banach spaces and
G X
-^
:
Z
X
G
(.To).
mapping. Suppose
be a vector-valued
continuously (Frechet) differentiable in the neighborhood of xq with the derivative denoted by
onto Z.
Then xq is said to be a regular point of
if G' {xq) maps
Lemma
X
2 Let
Z
and
Banach
be
Consider the maximization problem of
spaces.
P(«) = max/(a;)
(51)
subject to
<u
go {x)
(52)
and
G (x) <
where /
and
is
A'
:
— K
»
and go
the zero of the
the solution ai
u
=
:
X
^>
M.
are real-valued functions
Banach space Z. Suppose
0, xq, is
(53)
that f
a regular point. Let
fi
and
G X
:
—>
concave and go
is
any multiplier of
be
Z
is
is
a vector-valued
mapping
convex, and moreover that
Then
(52).
fi
is
a subgradient
ofP(O).
Proof. This lemma is a direct generalization of Proposition 6.5.8 of Bertsekas, Nedic and Ozdaglar
(2003, p. 382) to an infinite dimensional maximization problem.
Theorem
7 Let
X
and
Z
be
Banach
Consider the maximization problem of
spaces.
P (u) = max/ (x)
subject to
G(x) <
where /
:
X
— R
mapping and
>
is
is
+u
a real-valued concave function and
Z
the zero of the vector space
and u
is
and
G
:
X
—*
Z
is
a convex vector-valued
a perturbation. Suppose that Xq
Suppose also that xq is a regular point ofG and that f and
differentiahle in the neighborhood of xq. Then P(0) is difjerentiable.
to this program.
G
is
a solution
are continuously (Frechet)
Proof. From Lemma 2, it follows that if there is a unique multiphcr, P has a unique subgredient
and is thus differentiahle. Proposition 4.47 in Bonnans and Shapiro (2000) establishes that under a
weaker constraint qualification condition than regularity, this problem has a unique multiplier.
X
M
and
{X) to be the space of all measures defined on Borel sets of
{X)
is
nonnegative,
countably
additive, and has the property that ^ {X) = 1. Let
G
^
C {X) be the space of all bounded real- valued continuous functions on X. Then, following Parthasarathy
(1967, p. 40) we have that:
Next, consider a metric space
X
.
An element
M
Definition 7 The weak topology on
Foranyi
e
5{(/i,...,/fc;ei,...,£fc)
=
M (X)
M
(A')
is
,
the topology with the following sets as basis:
/i,
...,
A- 6
\u:veM[X),
46
C (X)
f,d^-
and
£i,
f.diy
...,£fc
€ IR+,
<ei fori =
l,...,k\.
)
.
The key
Theorem
result for us
X
8 Let
Theorem
is
from Parthasarathy (1967,
6.5
compact metric space, then
he a
M (X)
is
p.
45):
a compact metric space with the weak
topology.
X
Next, recall the Caratheodory's Theorenl, with conv {X) denoting the convex hull of
X e K^
(e.g.,
Theorem
some
for
Proposition 1.3.1 in Bertsekas, Nedic and Ozdaglar, 2003, pp. 37-38):
X
(Caratheodory's Theorem) Let
6 R'^ then any x e conv {X) can be represented as
..., Xm from X such that X2—xi,...,Xm. — x\ are linearly independent.
9
,
a convex combination of vectors xi,
Corollary
X6
Let
1
M.^
may
then any x G conv [X)
,
no more than
be represented by
N
-\-
vectors
\
ofX.
Suppose
Proof.
Theorem
would violate
not, this
—
independence of X2
linear
x\,...,Xm
—
a^j
as stated in
9
This theorem and
corollary imply that the convex hull of any subset of the A''-dimensional
its
+
Euclidean space can be achieved by A^
and
vectors,
1
will
be useful
in
reducing the dimension of
randomizations below.
Appendix B: Proofs
11
for Section 3
some public history /i*, then ^^ (/i') = 1, x[ (/i') =
cf yields utihty v [F [Kt (/i*"^) Lt (ft'))) to the government, which
is greater than its equilibrium payoff following ft*, given by the left-hand side of (8). This contradicts
sustainability and establishes that (8) is necessary in any sustainable mechanism.
Proof of
F
{Kt
Lemma
(/i'"^)
To
,
Lt
1:
If (8) is
and
(ft.*))
see that (8)
is
c[
violated following
=
,
is
5,
rij
M*
all
(i.e.,
G
x
>
tVI*, F'
and v
mapping that
Let p*
=
p*
if
(0)
G Q, K[j^i €
achieve this objective.
the
Q,
=
0),
we only need
R+ and
Cj
€
to
always preferred. Since from Assumption
show that v^
The
Ct-
is
M*
\
{K[j^i, Cj
|
M* ) =
is
achievable for
following simple combination of strategies would
Also denote c[ = c!^ be
consumption to all individuals irrespective of past and current reports.
^t-s (ft*"'^) and Mi-s = Mt-s for all s > 0. Then the following strategy
allocates zero
=
Xi-s (ft*~*)
which means that
7^
c'^
Let p' be the history of actions by the government.
combination would ensure
if p*"-'
K[^-^,
(
equivalent to relaxing the constraint on problem (4), so
>
reducing v^
sufficient for the best sustainable inechanism, note that
each citizen
for
=
p'~^, then al
v^ lKf_^_i,c[
\
i
M*
)
and
=
for all
for all
t,
t:
(1) for
we have that
the citizens,
if
a®; (2) for the government, F, such that
"
p*
=
p*~^
if
a=
p
,
(d
|
a'^), for
then a]
= p*~\
=
some
a,
and
then F involves
= at (ft*), Mt = Mt, and ^^{h*) = for all
€ if*; and if p*-^ =^ p*~\ then it involves
= 1, x[ (ft*) = F {Kt (ft*-^) Lt (ft*)) for all G H\ and cj = cf
We next need to show that these strategies are sequentially rational. Consider the citizens first; it
suffices to note that following a history where p*~^ ^ P*""^, the government is playing
(ft*+^) = 1,
(/^'"^') = F {Kt+s {h^+'~^)
Lt+s (ft*+')) and 5^+, = cf+,for all s > 0. Therefore, any strategy
x't+s
ft
(ft*)
ft*
^t (ft*)
ft*
,
(Jt_,_^
,
other than q® will give
type history
6* in
some
utility less
than
the current period, which
proves that this strategy
is
=
for the
1,
x't+, (ft*+^)
government starting
for all
M*
G
M*
,
A't+i G
E [X^^o /^*'" (^j
|
=F
cj
£
\
^*,ft*]
to
an individual with
It is also
sequentially rational for the
P*"^> there will be no future output to expropriate, thus
{Kt+s {h'+'-') Lt+s (ft*+^)) and 4+, = cf+, is a best response
^
,
in all of its information sets for all s
K+ and
Ot+s)
the utility that always playing a® delivers. This argument
sequentially rational for the citizens.
government, since after any history of p*~^
playing ^t+, {h*+')
is
Ct,
>
0.
The
fact that vf
(
K'^^ i cj
,
|
M* =
)
then implies that the best deviation for the government
47
is
also Ct {h*)
=
l,x't (/i*)
= F [Kt
(h^'^)
,
U
(/i*))
,
c[
=
cf and K[^^
(/i*)
=
G
for all h*
iJ* establishing
that (7) takes the form (8).H
Proof of Theorem 1: Take equilibrium strategy profiles V and a that support a sustainable
for all t and /i' e iJ*, and from Lemma 1, (8) is satisfied.
mechanism. Then by definition £,t{h*) =
Let the best response of type 6* at time t according to a be to announce Ztx (^*, /i*""') given a history
of reports zf"^ (^*"^,/i*~^)
4
(e^*,/!*-!)
= (4"^
and public history
mechanism given history
a best response, we have
this
ii
[4
Now
^',r,/i*-i]
{9',h'~')
>
/i*~^
{i
I
Let
/i*~^.
(l9*"^/l*''^) ,zt,r {0\h*)).
Denote the expected
be u [4 (^*,^'"^)
(0',/i'-i)
[4
By
6'*,r,/i*~^].
|
e*,r,/i*-i]
utility of this individual
under
definition of zf
being
4 (0',/i*-i)
for all
I
(61*,
/i*"^)
e Z* and h''' e
H'-\
consider the alternative strategy profile for the government F*, which induces the action profile
UMt,^tih'),Xt{h^),x[{h*),c[\°°
1
such that
=
a direct submechanism) and c[r* ,a*,h]
^, (ft*)
=
for all
c[r,a,h], L[T* ,a*
,h]
t
and
/i*
G //*,
= L{T,a,h],
M* = M; (where M;
and x [T,a,h]
=
is
x [T* ,a*,fi\.
Therefore, by construction,
w[0*,/i*-i |0*,r*,/i*-i]
=
w[4(6l',ft'-i) I^Sr,/!*-^]
(54)
= (a* a') is a best response along
and government strategy profile F*.
Moreover, by construction, the resulting allocation when individuals play a* = (q* a') against F* is
the same as when they play a against F. Therefore, by the definition of F being sustainable, we have
F >Za r' for all F' G Q. Now choose a' to be identical to a oflf-the-equilibrium path, which implies that
F* >Za' r' for all F' 6 ^ or that (8) is satisfied, thus establishing that (F*,q*) is an equilibrium.
for all 9
£ Q' and
all ft.*~^
£ H'~^. Equation (54) implies that a*
the equilibrium path for the agents against the mechanism
\
M*
|
Appendix C: Proofs
12
In this appendix,
for Section 5
we provide and prove a number of results used in the anatysis of Section 5, ultimate
Theorem 3. Throughout this section we assume that Assumptions 1-3
the building up to the proof of
and
5 are satisfied.
12.1
Properties of the Function U{C,L)
As mentioned
in the text,
to establish the concavity of
(1984a, 1984b) and allow for stochastic mechanisms,
specified in footnote 16.
lem.
Recall that
U{C,L)
is
i.e.,
U{C,L), we follow Prescott and Townsend
Mt = (ctJA
Z* x H^
^ A (E+
x
[O,/]) as
a solution to a finite-dimensional maximization prob-
Moreover, using the single-crossing property (Assumption
2),
we can reduce the
compatibility constraints to only the constraints for the neighboring types, thus to
A'^
static incentive
constraints (there
no incentive compatibility constraint for the lowest type, ^q). In addition, there are the resource
sum of consumption and labor supply levels. Recall also that only (C, L) 6 A will
enable this maximization, program to be well defined by making the constraint set non-empty.
< Z < [} be the set of possible consumption-labor allocations
< c < c,
Let C = {(c, /) £ R^
for agents. Let V be the space oi
+ 1-tupIes of probability measures on Borel subsets of C. Thus
each element (^ = [C(^o), •••,C(^a')] in ^ consists of
+ I probability measures for each type di G 9.
Let us also denote the fraction of individuals with type 6 at any point in time by n {9), where clearly
is
constraints on the
:
N
N
48
Then
the quasi-Mirrlees problem can be defined in the following
max
[/(C(/z*),L(/i'))=
way
Y^n{e) f u(c,l-e)C{d(c,l),e
\
h*)
(55)
eee
subject to
u(c,
I
I
OiXidic,
I),
>
Oil h^)
u{c,
/
0)C(d(c,
I
I
Y,TT{e)
I
/),
for
some
i
=
1,
...,
A^
(56)
I
c(:{d[c,l),e\h')<C[h})
Y^ tt{9) I iad{c, I), e\h*)>L
eee
h}) for all
e,^i
(57)
{h')
(58)
•'
(C, L) £ A.
Before deriving properties of the function U{C, L),
(58) define the constraint
Lemma
3 The solution
Proof.
The proof
we need
to ensure regularity. Let (56), (57)
and
mapping.
to (55) is a regular -point of the constraint
from single-crossing property (Assumption 2), all
and also linearly
independent from (57) and (58), thus the constraint mapping has full rank, A'' + 2, and is thus onto.
Our main result on the function U{C, L) is:
follows
from the
mapping.
fact that
incentive compatibility constraints in (56) are linearly independent from each other,
Lemma
in
4 U{C, L)
L and
is
and concave on A, nondecreasing in
well-defined, continuous
C
and nonincreasing
dijferentiable in {C,L).
we show that U[C,L) is well-defined, i.e., a solution exists. For this, endow the
V with the weak topology. Since C is a compact subset of K^, Theorem
8 above establishes that V is compact in the weak topology, and the constraint set is compact in the
weak topology as well. Moreover, the objective function is continuous in any ( G V, thus establishing
Proof.
First,
set of probability
measures
existence.
Next, to show that U{C, L)
is
continuous, note that with the lotteries the constraint set
Then from Berge's Maximum Theorem
U(C,L) is continuous in (C,L).
(e.g.,
Stokey, Lucas and Prescott, 1989,
Theorem
is
convex.
3.6, p.
62),
Concavity then follows from the convexity of the constraint set and the fact that the objective
is concave in ^ € P.
function
U{C, L)
is
also clearly nondecreasing in C, since a higher
in L, since a higher
L
C relaxes
constraint (57) and nonincreasing
tightens constraint (58).
Finally, to prove differentiability, note that the regularity condition is satisfied from Lemma 3 and
moreover, the objective function in (55) is continuously diflerentiable at all points of the constraint set.
We can therefore apply Theorem 7 using the strong topology, establishing that U{C, L) is differentiable
in (C, L). This completes the proof of the lemma.
The necessary
Lemma
5
A
is
properties of the set
(1
are derived in the next
lemma.
compact and convex.
Convexity: Consider {C° L°) [C^ L^) G A and some C°,C^ feasible for (C°,L°) and
a e (0, 1) C" = qC° + (1 - a)C^ is feasible for (aC° + (1 - a)C\aL° +
— a)L^), so that this set is non-empty. Moreover, since C°iC^ satisfy (56), (57) and (58), C" satisfies
Proof.
(CSL^)
all
A
,
,
,
respectively For any
three of these constraints, establishing convexity.
49
.
Compactness: A
(CjL") 6
(C°°,L°°).
since
V
is
is
clearly
bounded, so we only have to show that
Since this sequence
is
in a
bounded
set,
We just need to show that {C°°,L°°) e A. Let C" be a feasible element
compact under the weak topology, C" — C°° € V, which imphes that C°°
is
A
feasible for {C°°,L°°), therefore
w=
(C",L"), and
some sequence x
= {xt}^Q
as
y^^6^v{xt)
W
the set of feasible promised utilities
for
satisfies (56)-(58)
closed.
is
define a promised utility for the government for
Then
it is closed. Take a sequence
has a convergent subsequence, {C'^,L^) —*
it
>
and so C°°
Now
A.
is
defined as
oo
W = {w
3x e R°°
:
s.t.
for
any
t
there
is
some L
s.t.
(L
—
xt,
L) G A,
w = Y^ 5'v{xt)}
t=o
Lemma
6
W=
[0,-u}].
Proof. Since v{0)
=
To
that
0, it is clear
element. Moreover, clearly any
w
<w
is
further analyze the best sustainable mechanism,
sively as in equations (27)-(30) in the text.
Lemma
(27)- (30)
arg
the minimal element.
is
also achievable, so
The
let
following
By
definition iv
W must take the form
is
the maximal
[0,w].
us rewrite the maximization problem recur-
lemma
is
immediate:
7 The solution to (24) subject to (23) and (25) is equivalent to the solution to the program
combined with a choice of initial promised value to the government, wq, such that wq =
max„gw V (w)
Proof. The proof follows from Thomas and Worrall (1990). Clearly
anj' solution to (27)-(30) gives
a sustainable mechanism. Moreover, the ex ante utility for the citizens from any sustainable mechanism
can be obtained as V (w) from (27)- (30) by an argument analogous to the principle of optimality (see,
Stokey, Lucas and Prescott, 1989). It then follows that V (wq) = max^jgw ^ (^f) gives the best
sustainable mechanism.
Next note that the constraint set in the program (27)-(30) is not convex, and randomizations
over the current consumption and the continuation value of the government may further improve the
value of the program (which is the reason why we introduced histories /i*). So analogously to the
quasi-Mirrlees problem, we now consider randomizations. Now let q = {C,L,x,w') € K*, C [w) =
{q e R^ (27)-(30) are satisfied for given w], and let Z be the set of Borel subsets of C {w) Then let
the triple (C (w) Z, p.) be a probability space. Let V (w) be the space of probability measures on C (w)
endowed with the weak topology. Incorporating randomization, we can write the recursive formulation
e.g.,
:
,
Problem Al
V(w)= max [\U{C,L)+pViw')]ndq)
(59)
iev{w) J
subject to
C+x
v{x)
+
Sw'
w=
< L
>
^-almost-surely
(60)
v{L) ^-almost-surely
f[v{x)
+
(61)
6w']^{dq)
(62)
and
(C, L)
6 A and w'
eW ^-almost-surcly.
Note that the resulting solution to this program
and {wt (/i*)}^o ^^ specified in the text.
will
50
correspond to stochastic sequences
(63)
{.-r^
(ft.*)}(_Q
1
Lemma
V {ui)
8
is
concave.
,
Proof. Consider any wq and wi and ^q and ^^ that are the solution to tlie maximization problem.
w — awo + (1 — a)wi for some a 6 (0, 1). Let ^„ = a^Q + (1 — a)si- Constraints (60) and
hold
state by state, and are satisfied for both ^g ^^d (,i, and therefore must be satisfied for ^^.
(61)
Consider
Constraint (62)
is
linear in ^, therefore
we have Viawo +
linear in ^„,
(1
^^ also
satisfies this constraint.
— a)wi) > aV{wo) +
(1
— a)V{wi),
Since the objective function
is
.
establishing the concavity of V.
The above lemma establishes the concavity of V using arbitrary randomizations in the maximization
(59) The next lemma shows that a particularly simple form of randomization is sufficient to
problem
.
achieve the
Lemma
maximum
9 There
of (59).
exists ^
is
G
P [w)
achieving the value
and {Ci,Li,xi,w'i) with
points, {CojLojXqjW'q)
V (w)
probabilities
with randomization between at most two
andl —
£,q
^Q.
Proof. To achieve convexity, we only need the constraint set to be convex. The constraint set here
C {w) G M*. Recall Theorem 9 and its corollary, which imply that the convex hull of C (w) can be
achieved with 5 points.
Suppose, to obtain a contradiction, that there are more than two points with positive probability.
consider a case of three points, since the same argument applies to any finite number of points.
Suppose that randomization occurs between (Co,Lo,xo,Wq), {Ci,Li,xi,w[) and ((72,1-2, 2:2, 1^2) with
probabilities Co'^1,^2 > 0- Suppose without loss of generality that v{xo) + 6wq < v{x2) + Sw'2 <
We
+ 5w[
v{xi)
and
let
a €
[0, 1]
be such that v{x2) +6w'2
—
a{v{xo)
+ 3wq) + {1 — a){v{xi) + 5w'i).
Suppose
first
t/(C2, L2)
+ (3V{w'2) >
a[U{Co, Lo) + pV{w'o)]
+
(1
- a)[U{Ci,Li) + pV(w[)].
Then an alternative element ( e
{w) assigning probability I2 = 1 to {C2,L2,X2,wi,) is feasible
yields higher utility than the original randomization, yielding a contradiction. Next suppose that
V
UiC2,L2)+l3V{w'2)<a[U{Co,Lo)+0Viw'o)]
Now
consider an alternative ^ e
ity ^1
+
(1
~
Q;)C2 to
V [w)
+
and
{l-a)[U{Ci,Li)+l3V{w'j)].
assigning probabihty ^q + ce(,2 to (Cq, Lo,xo,Wq) and probabilis again feasible and gives a higher utility than original
{Ci,Li,xi,w[), which
randomization, once again yielding a contradiction. Therefore, ^ must satisfy
+
U{C2,L2)
PViw'2)
=
+ {l-cx)[U{Ci,L^)+pV{w[)].
[U{CQ,Lo)+l3V{w'o)]
But then the optimum can be achieved by simply randomizing between (Co, Lq, xo, w'o) ^^^d (Ci Li,xi, Wj)
+ q^j to (Co, Lo,xo,Wq) and probability $1 + (1 — a)^2Lemma 9 implies that we can focus on randomizations between two points. We denote solution for
any w by Ci{w) Li{w) Xi{w) ,w^{w) ^^{w) for i £ {0, 1}, and rewrite Problem Al in equivalent form:
Problem A2:
,
with probabiUties ^g
,
,
,
V{w)=
max
" " "
L^»'
;^e,[[/(C,,L,)+/3^K)]
1
J
(64)
t=o.i 1=0,1
subject to
Ci+Xi<
v(xi)
Li for
i
=
0,
+ 5w[>v(Li)ioTi =
w=J2^^ [^(^^) +
(65)
0,1
'^^'•]
(66)
(67)
i=0,l
(C,:,
Li) £
A
for
I
=
51
0, 1
and w' G W.
(68)'
—
1
in the text, the fact that at every date, there is randomization only between
two points also implies that the aggregate public history can be taken as /i* e {0, 1}
Next we would like to establish that V [w) is differentiable. This does not follow from Theorem 7,
since V [w) includes the term V{'w[), which may not be differentiable. Instead, we can apply an argument
similar to that of Benveniste and Scheinkman (1979) to prove differentiability (see also Stokey, Lucas
and Prescott, 1989).
Returning to the notation
.
Lemma
10
V (w)
is
differentiable.
Proof. From Lemma 9, in Problem Al when w = wq, the optimal value can be achieved by
randomizing between wj (wq) for i = 0, 1 with probabilities pi. Let V{w' {wq)) = Y^^-q j PiV{w', (wo))
and w' {wo) = X!^,;=o iPi''^i {'^o))- Then consider the maximization problem
Wiw)= max
f\U{C,L)+pV{w'(wo))]^{dq)
(69)
subject to (60), (61) and
w=
which only
By
from Problem Al
differs
in that
Lemma 8,
proof of Lemma
the same argument as in
(69) does not affect the
+
[v{x)
5w'
{wo)](. {dq)
,
V{w') and w' are held constant at V{w' {wq)) and w' (wq).
is concave (the fact that we have V (w' (wq)) fixed in
Next the same arguments as in Lemma 9 establishes that
W (w)
8).
W {w) can be equivalently characterized by the following maximization problem:
Problem A3:
W(w) =
^ max^
^^
{C;,Ci,L.,I,}.^„j
Yl
^'
[U{Q,L,) + l3V{w'{wo))]
j=0,l
v{xi)
Ci+Xi< L,
+ 6w (wo) >
for
2
=
0,1
v{Li) for
i
(70)
=
0,
(71)
thejr don't
1=0,1
(Q, Li) e A
Since
W [w)
constant here, and
all
Theorem
?
=
0, 1
and w' € W.
(73)
imphes that it is also differentiable
other terms are differentiable. Moreover, we have
concave,
is
for
7
pV{'w' {wq))
is
just a
W {w) < V {w)
(74)
W {wo) = V (wo)
(75)
and
by construction.
Prom Lemma
8,
V (wq)
is
concave, and therefore
In particular,
subdifferentials.
if
/
is
—y
is
convex. Convex functions have well-defined
convex, there exists a closed, convex and nonempty set
df such
v ^ df and any x and x', we have f {x') — f (x) > v{x' — x) (see Rockafellar, 1970, Chapter
23 or Bertsekas, Nedic and Ozdaglar, 2003, Chapter 4). Let —dV{w) be the set of subdifferentials of
— y, i.e., all —1/ such that —V{w) + V{w) > —v-{w — w). By definition, —dV{w) is a closed, convex
that for
all
and nonempty
set.
Consequently, for any subgradient
— i^of —dV
iy{w-wo)>V{w)-V (wo) >
where the
(75).
is
first
we have
W (w) - W [wq)
,
by the definition of a subgradient, and the second follows from (74) and
{wq))
also a subgradients of
(wq)- But since
(tuo) (and thus
—i/ must be unique, therefore —V (wq) (and thus V {wo)) is also differentiable.
inequality
is
This implies that -i/
differentiable,
[wo),
—W
is
52
W
—W
Proof of Theorem 3
12.2
V
Proof. Since
is
differentiable
Lemma
from
4 and concave from
Lemma
8,
the first-order conditions
are necessary and sufficient for the maximization (64). Assign the multipHers ^^Ki to the constraints in
(65),
^ji/jj
and 7 to constraint
to those in (66)
and
(67),
let
V [w) be the derivative of V {w) at w, we
have
with both equations holding as equality for
6lntW. Therefore,
w'^
^V {w[)
again with equality for
< -^i -
V
GintW. Moreover, since
w'^
is
(76)
-y,
differentiable,
we have
V (w) > -7
again with equality for
w
£lntW.
In addition, combining first-order conditions
Part
1.
establish this part of the theorem,
=
=
7
at
t
=
suflBces to
it
L) GlntA,
for {C,
iU) fori
4<iv'
V [w),
wq maximizes
First note that the initial value
V {wq)
and
To
1:
we have that
+ Ul[C„L,) =
Uc{Ci,Li)
or
(77)
and
= 0,L
show that
since
V {)
Suppose, to obtain a contradiction, that
0.
pV
This implies from (76) that
[w^) /5
Wo for all t, and (66) never binds. This
1.
=
so that wl
0,
=
is
(78)
i/)^
=
V',:
>
at
t
=
for
i
=
differentiable, this implies
at
t
=
for
both
i
=
wq. Repeating this argument yields
is in turn only consistent with xt =
for all t, which
wt =
then implies Ct = Lt =
for all t. This cannot be optimal, however, since i/'j =
both z =
and 1
implies Uc(Ci, Lj) -I- UL{Ci,Li) = 0, which does not have Cj = Lj =
as a solution (this follows from
Assumption 6, since otherwise w would be equal to zero, and the inequality in Assumption 6 could not
be strict). This yields a contradiction and establishes that 0i > for z = or 1 at time i = 0, so initial
(C, L) cannot undistorted, and from Proposition 2, there is a positive marginal tax rate on even the
highest type.
Part
Let
2:
/i*
e
{0, 1}*,
=
i
and
0, 1
fix
V {w[) <
Combining
this
with (77) and
(/',
>
w
some
e IntW. Since
-7
-t/)^
V
then implies that
i
=
<
(5
and
V {w) <
0,
(76) implies
0, 1.
yields:
V (w) > V {w[)
Concavity of
for
/3
uil
> w
ior
i
for
=
i
0, 1,
=
0,
L
establishing the claim that the sequence
nondecreasing. Since each Wt (ft.*) is in the compact set [0,w], the stochastic sequence
{wt (/i*)},_o
This im{wt (/i*)}^Q must converge almost surely to some point w* meaning p\imwt{h*) = w*
mediately implies {xt (ft')},=o almost surely converges to some x* (i.e., plimxt (/i*) = x*), and also
is
,
.
plim Ct (/i*) = C* and phmL^ (/i*) = L* are feasible (given plimxt (ft*) = a;*) and are optimal from
the concavity of U {C,L), this is a solution to the maximization in (64), establishing the existence of a
steady state as claimed in the theorem.
Recall from the above argument that {wt
w* < w. Then we must have
only possible
if ip*
=
for
i
V [w*)
=
<
-ip*
0, 1 (recall
(ft*)} T
-
that
7* for
>
ipi
''^*
i
=
almost surely. First suppose that P = 5 and
(w*) > -7*, which is
0, 1 and from (77),
0), establishing
V
the claim that the sustainability
become slack asymptotically. This immediately implies that, asymptotically, the
solution to problem (17)-(18) with {Xt (ft*)}^o "^ ^* ^^'^ ^^^ solution to problem (19)-(20)-(12) coincide
as required for an asymptotically undistorted allocation as specified in Definition 4.
constraints, (66),
53
Second, suppose that
(5
=
5
=
and w*
w. Recall that
w=
m.ax.(^c,L)eA
"^
(C, L) be a solution to this program satisfying the condition in Assumption
v{x)/{l —
> v{L)-
6)
This implies that at (^C,L,x), the constraint (12)
i^^
is
x
^ ^)y ^nd let
= L — C, and
and
therefore, the
~ C)
/ {^
so that
6,
slack,
solution to problem (19)-(20)-(12) coincides with the solution to problem (17)-(18) with
that the best sustainable mechanism
Next consider the case
in
is
which P
»
x, so
asymptotically undistorted.
<
6.
Now we
have
^V {Wi,t+i) + A<V' iwt)
Recall that, as established above, {wi^t}
1
for
=
i
0, 1.
<
w*. To derive a contradiction, suppose that w*
V
PV
X {h') —
w. This
(w*) /5 + ip* <
(w*) for some ip* ^ 0, which is impossible in view of the fact that
{w") <
(unless Wt = wq for all t so that
{w*) = 0, which is ruled out by the argument
p < S and
in part 1). Therefore, we must have {wi,t} T w). The same argument as in the previous paragraph then
implies that
V
V
establishes that the best sustainable
Peirt 3: Suppose that
P >
5.
mechanism
asymptotically undistorted.
is
Then, {wt (^*)}
is
no longer nondecrcasing.
If
{wt (h*)} converges
some w* < w, then equation (34) in the text must hold as t ^ cxd and hm^^oo —Uc/Ul exists
and is strictly greater than 1 as claimed in the theorem. Next, suppose that {wt} does not converge.
Since it lies in a compact set, it has a convergent subsequence. Suppose that for all such convergent
subsequences ijj^ = Q for i = 0, 1, this would impl}^ convergence to a steady state since we would have
for i = 0, 1 and for all t, yielding a contradiction. Therefore, there must exist a convergent
0J t =
to
subsequence with 0,; > 0, so that lim sup —t/c/t^L >
asymptotically, completing the proof.
Appendix D: Proofs
13
In this appendix,
we provide the proofs
Properties of
13.1
Consequently, distortions do not disappear
1-
for Section 6
more general environment.
for the
U ({Ct, Lf }^o)
above proof, to show concavity and differentiabihty of W ({Cj, L(}^g), we introduce ranTo simplify notation, in this appendix, we suppress dependence on public histories /i*.
The original maximization problem without randomization is to maximize (9) subject to (10), (11),
and (12) as stated in Proposition 1. Recall also that Ot € 0, where 9 is at finite set (with A'^ + 1
0* — M-jelements). Therefore 0* for any i < oo is also a finite set. Consider next the functions q
—
0*
and /t
By definition, these functions assign values to a finite number of points in the set
[O, r|
As
in the
domizations.
:
>
:
0*
for
any
t
<
oo, thus
can simply be thought of as vectors of
I
where
Y =
[N [N +
for
1))
Xf e X(,
the set Xj
F(Y,1) <
+
2).
let
Xt
(i.e.,
Tt
oo.
now
ct {e^)
dG
<
{e*)
Therefore, Xt
Let X( be
(i)
Y, Kt+i <
=
{ct [6*)
,1
Y
(A''
(TV
and Xt
{6*)
,
+
1))
dimension. Moreover
< Y,
Kt+i,Xt}
.
(79)
is
a vector (of dimension
the set of all such vectors that satisfy the inequalities in (79),
and
denote the ith component of this vector, and Tt be the dimension of vectors
= {N (N +
distance metric, dt (Xt, X')
Let us
»
.
1))
+2). Xj
= (yj=i
i-^'t (»)
is
in
a compact metric space space with the usual Euclidean
- ^t
ii)f)
construct the product space of the Xj's
CO
(=1
54
Clearly the sequence
{q
(^*) ,lt (^*)
,xt,I^t+i}t-Q i^ust belong to X.
subset of X, which satisfy (10), (11), and (12), denoted by X.
Now by Tychonoff's theorem (e.g., Dudley, 2002, Theorem 2.2.8),
In fact,
X
is
it
must belong
compact
in the
to the
product
X
is a closed subset of X, and therefore it
topology. Since (10), (11), and (12) are (weak) inequalities,
with the product topology is meterizable, with
is also compact in the product topology. Moreover,
X
the metric
d(X,X') = Y,'P'dt{Xt,Xi)
(80)
t=i
X=
{Xt}'^Q G X. This shows that X endowed with the product topology is
X. Prom Theorem 8, the set of probability measures defined over a compact
metric space is compact in the weak topology. This establishes that the set of probability measures
V°° defined over X is compact in the weak topology.
We are concerned not with all probability measures, but those that condition at ( on information
< c < c,
< < 1} he the set of possible consumptionrevealed up to t. Let C = {(c, /) G M^
for
some
£
lo
and
(0, 1)
a metric space, and so
is
I
:
labor allocations for agents, so that 7^°° defined above
the set of all probability measures over
e 0'~S let V [0*~^] be the space of
+ 1-tuples of probabihty
measures on Borel subsets of C for an individual with history of reports 9^^^. Thus each element
0*~'^)] in a P* [6**"^] consists of iV + 1 probabilities measures for
61*""^),
...,C{On
( (. e*"^) = [C(6lo
[^*~^]'
each type 9i given their past reports, 0*~\ and is thus closed. Consider P =
Now,
C°°.
each
for
i
N
€
N
6''"^
and
I
I
is
I
UtsNUe'ee'^*
a closed subset of P°°. Since a closed subset of a compact space
Theorem 2.2.2), V is compact in the weak topology.
which
is
<
is
compact
(e.g.,
Dudley, 2002,
shows that the objective function is continuous in the weak topolwe have a maximization problem over probability
measures in which the objective function is continuous in the weak topology, and the constraint set is
compact in the weak topology, and thus there exists a probability measure that reaches the maximum.
Given this result, the rest of the analysis parallels that of Theorem 3. The key lemma is a generalization of Lemma 4, which is stated here.
Finally, choosing
/?
in (80)
ogy. This establishes that including randomizations,
Lemma
11 W({Ct, L(}J^q) is continuous and concave on A°°, nondecreasing in Cs and nonincreasing
s and differentiable in {C't,Lt}^o-
in Ls for any
This lemma can be proved along the
we
are no longer able to prove
point as defined in Definition
Lemma
lines of
3.
Lemma 4,
Thus,
all
except that in this infinite-dimensional space
the proofs assume that the solution
is
at a regular
6.
Proof. The above argument established that in the problem of maximizing (9) subject to (10),
and (12) over probability measures, a maximum exists and U{{Ct, Lt}'^^) is therefore well defined.
To show concavity, consider (C°,L°) and {C^,L^) and corresponding CiC"""- We have
(11),
u[c,i\e)
a
[{u{c,
I;
9)
u{c,ue))C[d{c,i),e)
-
u(c,
/;
9))C,°{d[c,
/),
9)
+
{I
-
a)
I
(u{c,
I;
9)
-
u{c,
I;
e)X\d{c,
I),
6)
>
In a similar
feasible
and
it
way we can show that ("
same utility as a(f
gives the
satisfies (10),
u{9)
Next, note that the constraint set expands
if
-I-
(1
—
(11),
a)C^
and
(12), this
convex combination
Cg increases or Ls decreases for any
must be weakly increasing in Cs and weakly decreasing in L^.
Finally, Theorem 7 applies to this problem and implies that U{{Ct,Lt}^Q)
{Ct, Lt}'^Q, completing the proof.
Lemma
12 A°°
is
compact and convex.
55
is
u{9).
is
s,
therefore
U
differentiable in
Convexity: Consider {Ct,L/}^Q and {Cf,L{}^Q e A°° and some C^^C^ feasible for
Now for any a G (0, 1) C° = aC° + (1 — Q:)C^ is feasible for
—
so
that
this set is non-empty. Moreover, since ( ,(* satisfy (56),
{a{Ct,Lt}^Q + (1 q) {CjiLil^g),
Similarly,
satisfies
and (58).
satisfies
it
as
well.
(57)
C"
C"
Proof.
{C(, Lt}'^Q and {CI,L[}'^q respectively.
Compactness: For any sequence {Cf L'^}'^^ £ A°°, {Cf L^}^o "^ (CT. -^DJ^O' there exists a
sequence {C"}J^o corresponding to {C", .f/jj^^Q, such that C" —* C°°i satisfying (56)-(58) and feasibility,
therefore {C^ Lf^}^Q £ A°° is closed. Boundedness follows from boundedness of C and L. u
,
,
,
Proof of Theorem 4
13.2
The proof
Theorem 4
of
is
similar to that of
Theorem
3,
except that we do not use the recursive
formulation. Instead, we work directly with the sequence problem and the necessary conditions of the
sequence problem.
Suppose to obtain a contradiction that n^ (/i*) =
for all i >
and all /i'
(almost surely). Then, Xt (/i*) =
for all t and all /i*. But in this case, if Lt >
for any t, then
the government can improve by expropriating the entire output at t. Thus we must have Lt (/i') =0
for all t and all /i*. Since, by hypothesis, {Cj (/i*) Lj (/i*)}^o SlntA^ with Lt{h^) >
is feasible
and the associated {Ct (/i*) ,Lt (/i')},^o GlntA°°' necessarily gives higher ex ante utility to citizens than
for all t and all /i* cannot be optimal, establishing a
Lt [h*) — Ct (/i*) = 0, the plan with Lt (/i*) =
contradiction and proving Part 1 of the theorem.
Part 2: Take {Ct [h*-) Lt (/i*) Kt+i ('i*)}^o *° ^^ P^'^* °f ^^^ optimal mechanism. Suppose that
{Ct (/i*) ,Lt [h^) ,Kt+i (^*)}(^o (almost surely) converges to a hmit, (C*,L*,K*), and let x* = L* —
Part
Proof.
1:
,
,
,
C* -K*.
=sup{ g e [0,1]
p\ixnt-^oc Q~*U^^ = 0} defined in the theorem is
1. To sec this, recall that by hypothesis, a steady state exists, so that
{Ct (/i*) Lt (/i*) Kt+i (/i*)}J^o ~^ (C*,L*,K'), thus {Ct (/i*)}^q is in the space c of convergent infinite
sequences (rather than simply in the space of all bounded infinite sequences, ioo)- The dual of c is ^i,
We
by proving that
start
ip
:
well-defined and strictly less than
,
,
is, the space of sequences
{yt}f-o such that Ylu=o 12^*1 "^ °°- Since Uci is equal to the Lagrange
multiplier for the constraint (15), it lies in the dual space of {Ct (/i')}^o i^^^' ^-S-' Luenberger, 1969,
that
which implies that limi^oo^Ct == 0' hence ip < 1.
The rest of the proof of Part 2 will consist of two cases.
Case 1: Suppose that {Ct (/i*) L* (ft')}J^o -^ (C*,L*) eIntA°° and that tp = 5. Then (40) and
(41) are necessary conditions for optimalitj'. Rearranging these equations and substituting for Uc,, we
have
Chapter
9),
thus in
£i,
,
^l,
_,
{Ht-f^t-i)v'{F{K\L*)))
WcFlAK\L')
^
f^yi^*)
'
and
Fk,,AK*,L')Wc,^^
where
all
{^,^,-f^,)v'{F{K\L*))FK,,AK*,Ln
^
(C ,L* ,K*).
{C*,L*) eIntA°°, equation (44) also holds as
derivatives are evaluated at the limit
Since {Ct
(/i*)
,
^
Lt (/i*)}^o
t
^
cxs
and implies that,
we have
Since as
(83) can
t
-^ oo, a steady state {C* ,L* ,K* ,x*) exists by hypothesis and
H
Since
l^t+i
~
Uq
is
proportional to
(/3*,
be written as
(p
l-h\
=
~*
5
and
^'^d
,( ,,
v'[x*)
=
^t
<
Mt+i
since, as established above,
fXt
^>
f^*
^
(0,
=
<^
,t+i ,, ,,
S
v'{x*)
<
1,
oo) (where the fact that
56
ast^oo.
we have that
p,*
>
follows
(84)
(84) implies that as
from Part
1,
i
—
since Ht+i
»
>
oo,
Mt
.
for some t).
Therefore, (^j — fl^__l)/|I^ —> 0, and from (81) and (82), we have that
>
-UlJUc.Fl, and FK,+MCi+J^Ct almost surely converge to 1, thus {C( (^*) Lj (/i') /\(+i (/i')}j'^o
must be asymptotically undistorted. The rest of the argument parallels the proof of Part 2 of Theorem
and
ij,f
,
,
1 where {Ct {h*),Lt (/i')}J^o ^_(C*,L*) €lntA^ and ip = S.
Suppose that either {Ct (/i*) L* {h')}Zo
(C*. L") eBdA°= (and f = d)or^<S.
First consider the subcase where tp < S. Then, equation (83) cannot apply. Since this equation must
hold for all {Ct (/i*) L* {h^)}Zo eIntA°°, we must have that {Ct (/i') Lt {h*)}Zo -" {C\L') eBdA°°.
This implies that we converge to some {C4(/i*),L((/i*)}~o -^ {C*,L*) eBdA°° and {Ct (/i*) L* (/i*) ,ii't+i (/i*)}^o
[C* L* K*) (since by hypothesis a steady state exists), as in the first part of the hypothesis. Thus we
only have to consider the case where {C* ,L*) 6BdA°° with and ^p <5.
Now let us turn to the subcase where [C ,L*) GBdA^ (and ^p < 5). Suppose also that v (x*) / \\ — 5) <
w, where w is as defined in Assumption 7. Now (40) and (41) no longer appl}-, but since v (a;*) / (1 — 6) <
w, the first-order necessary conditions with respect to Xt and Kt+i imply:
3.
This establishes the result for Case
Case
2:
^
,
,
,
,
,
,
5^itv'{xt)
-^'(/it+i
where
recall that k;
eliminating Kt,
we
- t^tW iF{Kt+i,Lt+i))
-
S*Kt
=
Fk,+^ +5^^^Kt+iFK,+, -S*Kt
=
0,
the multiplier on the resource constraint (39). Rearranging these equations by
is
obtain:
i^--'iFiK..uLt,.))
.P.:y.,.-) -^'(^(^^^*))
v'(xt+i)-v'{F{Kt+i,Lt+i))
v'{x')-v'[F[K*,L*))
f.t,^
/i,
'
^
'
where the second equality evaluates the expression at the limit [C* ,L* ,K*,x*). Now to obtain a
contradiction, suppose that as t — oo, we have /Xj — oo, and thus Ht+i/fJ^t > 1- Inspection of (85),
combined with the fact that v' (xt+i) — v' {F{Kt+i,Lt+i)) >
(since F{Kt+i,Lt+i) > Xt+i and v{-)
is concave), implies that this is only possible if 5Fx,^i {K* ,L*) < 1. However, we have Uci < ^Ct+r
Fxt+it thus as t —t 00, ^pF^^^-^ > (5Fa',+i > 1, where the inequality follows from the assumption that
ip > 5 and contradicts SFk^+i {K*,L*) < 1. Consequently, we have ^^
/x* < cxd at the limit point
[C* ,L* ,K* ,x*), so that (12) is slack, and therefore the solution to problem (19)-(20) coincides with
the solution to problem (17)-(18) with
[h}) — x*
Finally suppose that (C* ,L*) €BdA°° but also v (x*) / {1 — 5) = w. Assumption 7 then implies that
(12) is slack, and once again the solution to problem (19)-(20) coincides with the solution to problem
[h^)
X*. This completes the proof of Case 2 and thus of Part 2.
(17)- (18) with
Part 3: Suppose that ip > 5. In this case, (83) implies that Uq is proportional to
> 5 as t — oo.
This implies that (yu^ — ;Li(_i)//i( >
as i -^ oo, so from (81) and (82), aggregate distortions cannot
>
»
^
X
>
^
X
i/p
>
disappear, completing the proof.
Appendix
14
14.1
E: Proofs for Section 7
Proof of Theorem
The proof
of this
5:
theorem follows the structure of the proofs of
Lemma
1,
Theorem
1
and Proposition
1.
Proof. As
in the proof of
continuation play in which
a submechanism
s
>
0.
We
first
Mt
all
at time
t
Lemma
we
1,
first
need to show that there exists a sequentially rational
agents supply zero labor. Suppose that the government has announced
and has
capital stock Kt,
show that a deviation by an
individual,
i'
and a\^^
with type
— q^
9\
^
for all
9q to
i
€
[0, 1]
and
for all
some other strategy that
not profitable (as noted in footnote 10, we think of an individual with
positive measure e deviating, and take the Umit e — 0, since there is a continuum of agents). Without
involves supplying positive labor
is
»
the deviation,
i'
obtains utility u (0) / {\
—
(5)
(since
57
from Assumption
1',
x
(0
I
^)
=
for all
£
Q
and there
be no labor supply
will
for
This will generate output
F {Kt,£l'),
imagine the behavior of the government at the
since
all
imagine a deviation
=
by
other agents are supplying zero labor.
Now
to a message that corresponds to positive labor supply, say V
definition.
Now
any type in the continuation game).
last stage of
,
with X
(''
^t
I
> X
)
(o
|
the game, conditional on aj_^j
&J
)
= a®
for all
for all s > 1. Then the sequentially rational strategy of the government is to maximize
Ki+i = 0, since there will be no production in future periods. Consequently, the utilitymaximizing program of the government in the information set following the deviation is (suppressing
i
G
and
[0, 1]
(49) with
/i*-dependence to simplify notation):
max(l -
a) v [xt)
+a(
f[u{c[
-x
(z' (a* (0'))))
{h {z' {at {6')))
\
6^)]
dC
(0*)
Jc[ (2* {at {0*))) dG^ {6^) < F{Kt,el'), where recall that 2' {at {6^)) is the history of
reports up to time t by an individual of type 6* given strategy profile a. In view of Assumption \\ this
expression is concave in c for any strategy profile a, so the optimal policy for the government in this
information set is to redistribute consumption (what it does not consume itself) equally across agents,
i.e., c't (2* {at {d*))) = ct for all s* {at (^*)) £ 2*. This implies that as e —* 0, c^
0, and thus the devi-
+
subject to xt
^
ation payoff of
i' is
u
(0)
-x
(/'
0'')
|
+P
{u
(0)
-x
(o
|
6*'')) / (1
-
/3)
< (w
-x
(0)
(o
|
^'")) / (1
-
/?),
showing that a continuation strategy profile where all agents supply zero labor is sequentially rational.
Now consider two different types of deviations by the government. First, imagine the government
offers Mt ^ Mt, i.e., a different mechanism at the beginning of time t than the one implicitly agreed
in the social plan (M, x). Given the above-constructed continuation equilibrium, a\^^ = a^ for all
is a best response against this deviation.
Since maximal punishments are
i 6 [0, 1] and for all s >
optimal, a\^^ = a^ for all i 6 [0, 1] and for all s >
is optimal against this deviation, implying that
such a deviat'on would never be profitable for the government.
Second, u- before, the government can deviate at the last stage of time t. Again a\_^.^ = a^ for all
i G [0, 1] and jor all s > 1 is the maximal sequentially rational punishment against such a deviation.
Consequently, after any deviation by the government, there will not be any further production. Thus
the optimal deviation for the government involves K'tj^^ — 0, and again exploiting the concavity of the
government's continuation payoff in c, the sustainability constraint is equivalent to:
E,f^(5^
>
max
[(1
- a)v[xt+s)+a¥.t+s
{l-a)v
{xt)
+
a
/
(
[u
[u{ct)
(cj (0*)
-x
dG*
{h {z' {at {9')))
{9')]
for all
\
9,)]
dG'
(0*)
(86)
t.
Now, given an equilibrium pair of strategy profiles T and a, exactly the same argument as in
Theorem 1 implies that there exists another pair of equilibrium strategy profiles F* and
a* = [a* a') for some a' such that F* induces direct submechanisms. Consequently, we can write
the proof of
I
(86), in
terms of a direct mechanism, which gives
Finally, the
mechanism
is
same argument
(50).
as in the proof of Proposition 1 implies that the best sustainable
a solution to maximizing (9) subject to (10), (11), and the sustainability constraints of
the government given by (50).
14.2
Proof of Theorem 6
Proof. Suppose again that there are A'' -I- 1 types, i.e., Q = {9q,9i, ...,6is/}, ranked in ascending order
of skills, and with respective probabilities {ttctti, ...,7:^^}. Given the assumptions of the theorem (and
again suppressing /i*-dependence to simplify notation), we can write the program for the best sustainable
58
mechanism
as:
N
oo
subject to the constraints
,
oo
oo
J2P'
m)
[" (ct
-X
(It
> X^ /3*
e,)]
(ei)
I
[w (ct (e,-i))
- X {It
{9i-i)
I
(87)
^i)]
1/
for allz
=
l,...,Af,
(1
f;/3*+^
for all
-
a)
t;
(x(+,)
+
a f f^^i
[u
(q+, (^0) - X
j
I
I
> V
{Kt,
U)
(88)
and
t,
+
Xt+i
^
+
^,K
<F
(ct (e,))
1=0
and that
t,
The
Cj (^i)
>
for all
i
N
/
/V
a;*
for all
e,)]
(^t+. {0^)
I
Kt,
t
and
Sj
>
I
(89)
^i)
1=1
\
and
\
;^ mk {Bi
for all
/
t.
Given Theorem 5,
Assumption 2, implies that
we only need one constraint for each type other than the disabled type, Oq, where type i could deviate
to claim to be type i — 1. The second set of constraints, (88), one for each date, impose sustainability,
there
is
first set
of constraints, (87), ensure incentive compatibility for the citizens.
truthful revelation along the equilibrium path. This, together with
V {Kt,Lt) =
maxj;+cj<F{K-,,L,) (1 - o.)v{x[) + a Z^^lo '''j'" (^t (^i))i and finally, the
one for each date, impose the aggregate resource constraint.
As in subsection 6.1, we follow Marcet and Marimon (1998) and form the Lagrangian (see footnote
with the definition
last set of constraints,
41 for details):
N
oo
t=0
i=l
oo
^
AT
C
+ ^^Vt
i=0
(1
-
a)v{xt)
I
i
oo
t-Q
A'
r
Aj
is
>0
N
\
1=1
- xik
(^i)
|
Oi)]
\
=l
J
oo
oo
r
N
/
N
\>,
<=0
I
i=l
V
i=l
/J
the multiplier on the incentive-compatibility constraint of type
the resource constraint at time
Xt
/
[« (c^ {di))
tt=0
i=l
where
+a^7r,
for all
t
t,
and we have
left
the constraints that
?',
/3*77(
ct [6i)
is
>
the multiplier on
for all
i
and
t
and
implicit.
For Cf (9i) >
and Xt > 0, we can take first-order conditions, which, after canceling out the 0^
terms and defining Aq =
and Ajv+i = 0, yield;
(1
+ ant) TT^u
{ct (Oi))
+
(Ai
-
Ai+i) u'
(C( (5,))
59
-
Vt'^i
=
for all
i
=
0, ...A^
and
all t,
(90)
.
+
(1
+
(^,
-
[h
flMt) T^iX
-
m
+ {Xa'
Or)
I
-
/ij_i) tTjVl {I<t,Lt)
-
{fit
TitTTiFL
Recall that {^tj^o
di)
{Kt,Lt)
(xt)
=
^ nondecreasing sequence, thus
'^
First suppose that
I
- K+lX'
=
[k [Or)
for all
i
=
0,
Vk {Ku Lt) + ViFk [Ku U) - p-'vt-i =
t^t-i)
lit{l-a)v'
real Hue.
(0,)
{It
—
/ij
>
<
Tjt
for all
I
(91)
0,+l))
and
...A''
for all
all t,
(92)
t,
t.
(93)
possesses a unique limit point on the extended
it
then (90)-(93) establishes that there exists an allocation
and xt —* x*, in which distortions disappear eis claimed in the
/j,*
oo,
for all i,
with r^j — 77*, C( {9i) -^ c* >
theorem.
To complete the proof, we need to show that there does not exist any solution to the above maximization problem with /i^ — 00. To obtain a contradiction suppose that fj.^ — 00. Combine (90) for
and that A/v+i =
tj^pe A'' with (93) and using the fact that ij,^_^_i > fi^ >
(by definition), we have
>
»
that for
[d^)
all Ct
>
and xt >
-
(1
\
Both
fif
Next combine
u' {ct
fact that
that ct{9i)
have
=
disabled).
Ct {6,,) I
for all
>
{e,.))
-^
/j.^
(0t) i I*
It
(90)
cx)
i c* for all
become
t
a)I
0,
-
,,
\fl
^\
u'[Ct{eN))
(1
^
a.
sides of this equation are strictly positive
—* 00 implies that as
The
>
i
—y 00,
t
some
i
fif
and
<
z'
,,
]l' „
u'(c,
+ i(9n))
''
fact that Ht+i
^
Mt
—
>
a
0-
The hypothesis
that
fJ-t^i-
^
implies that as
=
for all
0,
i
...A''.
i
to obtain:
=
0, ...A^
and
i
—
»
From
1, ...A'^,
From Assumption
=
a)
\^'^'\ u' (c* (0.)) = W {c, [6,)) + i^llZ^llti)^' (c, (0,0)
+
i
by the
—
2
00,
\ci
{6i)
—
ct (^i')l
~^
0.
(95)
This argument then establishes
we must
would claim to have
the freedom of labor supply, this also implies that
since otherwise at
some point
and the resource constraint,
all 9i
/
^o
this also implies that as
t
—
00,
and
all
»
Xt i 0.
Now, suppose first that T = and c* =
t' for some t' < 00. We will show that
are reached in finite time,
this
i.e.,
Cf
(^,)
=
for all
cannot be part of a sustainable mechanism.
i
We
have
all t > t', ct {9,) = It {9i) =
for all i and xt = 0, so the continuation utihty of the government
— a)u (0) / {1 — 5) (since (0) = and x (0 ) = 0)- However, by hypothesis, ct'-i {9i) > for
at least some i, so there is positive output at t' — 1. Moreover, from (95), Ct'-i {fii) ^ Ct'-\ {9i') for
some i and i'. This implies that the government would prefer to deviate at i' — 1 to (Jj,_j = 1, and
that for
is
{I
i;
I
individuals (i.e., Cf^i {9i) = Cj/_2
government utility at i' — 1 (since
Ct'-i {9i) / Cf'-i {9i') for all i and i' in the original allocation), and its continuation utility from t'
onwards would still remain at (1 — o)u(0)/(l — 5). Since this argument applies for any t' > 0, it
and c* =
in finite time.
proves that tliere cannot be a sustainable mechanism that reaches Z* =
Hence, it must be the case that c* (0,) [
and It {9i) 0, but Cj {9i) > for all t. Then, combining (94)
and (95) implies that, as long as au' (0) ^ {I — a) v' (0), (j,t_^.i — fit i 0, contradicting Ht — 00, and thus
redistribute the output between xt
for all
i
and some
c*,_j).
and equal consumption across
This deviation
all
will necessarily increase
J.
>
establishing the theoremi.
60
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