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Technology
Department of Economics
Working Paper Series
Massachusetts
Institute of
Moral Hazard and Efficiency
with
in
Anonymous
General Equilibrium
Trading
Daron Acemoglu
Alp Simsek
Working Paper 10-8
April 27, 2010
RoomE52-251
50 Memorial Drive
Cambridge,
MA 021 42
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This paper can be
Social Science Research
(>'
Moral Hazard and Efficiency
General Equilibrium
in
with Anonymous Trading*
Daron Acenioglu^
Alp Simsek*
April 27, 2010
,
,
Abstract
A
"folk
theorem" originating, among others,
in the
work
of Stiglitz maintains that competitive
equihbria are always or "generically" inefficient (unless contracts directly specify consumption levels
anonymous markets). This paper critically
economy with moral hazard. We
first formalize this folk theorem.
Firms offer contracts to workers who choose an effort level that
is private information and that affects worker productivity.
To clarify the importance of trading in
anonymous markets, we introduce a monitoring partition such that employment contracts can specify
as in Prescott
and Townsend, thus bypassing trading
in
reevaluates these claims in the context of a general equilibrium
expenditures over subsets in the partition, but cannot regulate
among
We
the commodities within a subset.
not weakly separable)
how
this expenditure
is
subdivided
say that preferences are nonseparable (or more accurately,
when the marginal
rate of substitution across commodities within a subset in
and that preferences are weakly separable when there exists
no such subset. We prove that the equilibrium is always inefficient when a competitive equihbrium
allocation involves less than full insurance and preferences are nonseparable. This result appears to
support the conclusion of the above-mentioned folk theorem. Nevertheless, our main result highlights
its limitations. Most common-used preference structures do not satisfy the nonseparability condition.
We show that when preferences are weakly separable, competitive equilibria with moral hazard are
constrained optimal, in the sense that a social planner who can monitor all consumption levels cannot
improve over competitive allocations. Moreover, we establish e-optimality when there are only small
deviations from weak separability. These results suggest that considerable care is necessary in invoking
the folk theorem about the inefficiency of competitive equilibria with private information.
Keywords: competitive equilibrium, double deviations, efficiency, general equilibrium theory, mon-
the partition depends on the effort
level,
itoring partition, moral hazard.
JEL
Classification: D52, D61, D62,
"We thank Alberto
participants at
*
NYU
Bisin,
and
Glenn
MIT
Massachusetts Institute of
Ellison,
D82
Bengt Holmstrom, Robert Townsend, Muhamet Yilidiz and seminar
comments and suggestions.
Technology and Canadian Institute for Advanced Research, daron@mit.edu.
for useful
'Massachusetts Institute of Technology, alpstein@mit.edu.
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/moralhazardefficOOacem
Introduction
1
A
central question for economic theory
with complete markets, this question
is
equilibria are Pareto optimal
conclusively answered by the celebrated First
is
Second Welfare Theorems, which show
the efficiency of competitive markets. In economies
that,
under some regularity conditions, competitive
and every Pareto optimal allocation can be decentralized as a
competitive equilibrium. Nevertheless, the complete market benchmark does not cover
empirically-relevant economies where missing markets are ubiquitous.
missing markets in practice
important reason
for
know more about
their preferences, risks
on
sizable literature
and
is
many
Arguably the most
private information.
Individual agents
and actions than the market can observe. Despite a
economies with private information are
this topic, efficiency properties of
not yet fully understood. In this paper, we investigate the efficiency of competitive equilibria in
a subclass of economies wdth private information, those with moral hazard, where individuals
take privately-observed actions affecting their endowments (and/or production).
One approach
to the study of efficiency in moral hazard economies has been pioneered
by
Prescott and Townsend (1984a, 1984b). Prescott and Townsend propose the important idea
of considering insurance contracts as commodities that should also be priced in equilibrium.
Prescott and Towmsend show that competitive equihbria with moral hazard are (constrained)
Pareto optimal under two key assumptions: exclusivity and
that individuals can sign exclusive contracts and
employment
contracts.^
assumption,
full
We
monitoring,
is
more problematic.
is
kets."
buy
The
first
implies
a good starting point for the study of
Under
full
fully
The second
monitoring, contracts specify
for individuals in different states of nature.
imphes that firms or some other outside agency can
is
monitoring.
focus on exclusive contracts throughout the paper.
complete consumption bundles
This assumption
full
This essentially
monitor individual consumptions.
not only unrealistic but also goes against the spirit of "competitive mar-
Competitive markets should allow anonym.ous trading, so that individuals are able to
at least a subset of
commodities in anonymous markets without a central agency keeping
track of their exact transactions.
A
mous
systematic analysis of the structure and efficiency of competitive equilibria with anonytrading
is
not available, but a series of papers by Stiglitz and coauthors, most notably,
Greenwald and Stightz (1986), and
also
Arnott and Stightz (1986, 1990, 1991), claim that
competitive equilibria under these circumstances are always or "generically" Pareto subopti'
Exclusivity
insurance
is
may be
a less satisfactory assumption for insurance contracts,
also possible; see, e.g.,
Arnott and
Stiglitz (1991)
in
particular,
when informal
and Bisin and Guaitoli (2003).
Real-world
insurance and financial contracts often explicitly regulate what other contracts individuals can sign for the same
whether they can pledge the revenues of the same business. Exclusivity
context of employment contracts we focus on in this paper.
risks, or
is
much more
natural in the
mal. These claims are supported by local analysis of first-order conditions, though without a
rigorous proof that this type of local analysis
may
valid or economically important.'^
say that the inefficiency of competitive equilibria with
a folk theorem. This folk theorem
in applied work. It
and
is
credit
is
is
anonymous trading has emerged as
not only of theoretical interest but has been very influential
often invoked to argue that decentralized allocations in insurance, labor
markets are
inefficient
and necessitate government intervention
intuition for specific models in which this
(or to provide the
the case).
is
we consider a general equilibrium environment where the structure and
In this paper,
efficiency of competitive equilibria with
consists of a large
Hence one
number
of firms
anonymous trading can be
studied.
The economy
and risk-averse individuals. Individuals accept employment
contracts from firms and choose an effort level, which determines the probability distribution
over a vector of production.
economy
Individual effort
private information.
Commodities
in this
are partitioned, such that expenditures over subsets in a given monitoring partition
of commodities are observable (for example,
determined but not how this spenchng
Employment
resort).
is
is
how much an
individual spends on vacation can be
distributed across different activities in the vacation
contracts specify payments to workers and expenditure levels over the
subsets in the monitoring partitions as a function of the realization of the state of nature.
Prescott-Townsend economy
a singleton.'^ After
all
is
The
a special case where each subset in the monitoring partition
uncertainty
is
is
resolved (the underlying states of the world are reahzed),
individuals allocate the contractually-specified expenditures within the subsets in the partition
at given
market
prices.
We establish the existence of a competitive equilibrium
lem that characterizes equilibrium allocations (Theorem
ize the
if
above-mentioned
folk
theorem.
We say
1
that there
and an
indirect
and Proposition
is
no
full
maximization prob-
We then formal-
1).
insurance at an equilibrium
the marginal rate of substitution of some good between any two states
is
not one. can we
say that preferences are nonseparable (or more accurately, "not weakly separable"),
if
there
is
a subset in the monitoring partition such that the marginal rate of substitution between the
goods in the subset change
if
the effort level
is
modified. Conversely,
we say that preferences
'This is particularly concerning for three reasons. First and more importantly, this local analysis assumes
that a range of Lagrange multipliers exist and are strictly positive, though there is no mathematical or economic
reason for them to be so. One of our main results will establish the (constrained) optimality of competitive
equilibria under certain conditions explained below, thus invalidating this line of analysis.
Second, the local
assumptions and the first-order approach, which do not generally apply in
these environments (see Grossman and Hart, 1983, Rogerson, 1985, Jewitt, 1988 on the first-order approach).
Third, as we will show, it may well be that even if efficiency is "nongeneric," small deviations from this efficiency
benchmark might still lead to allocations that are approximately (e-) efficient and many inefficiencies identified
via this method may not be first order.
Another special case is one in which the partition consists of a number of singleton elements, which correspond to "monitored goods," and the remainder, which comprises "nonmonitored goods."
analysis
makes use of
differentiability
when
are weakly separable
there exists no such subset (this
standard separability assumptions often adopted
Theorem
result (contained in
2)
in theoretical
in the sense that a social planner
who
is
who
is
Note that
this
theorem
and applied work). Our
is
first
is
no
constrained suboptimal (inefficient),
is
constrained by the same moral hazard problems (but
allowed to monitor expenditures on
cation.'*
weaker than the
significantly
shows that when preferences are nonseparable and there
insurance at an equilibrium, then this equilibrium
full
is
silent
all
goods) can improve over the equilibrium allo-
on whether a
social planner
who
is
also constrained
by the same monitoring technology can implement such a Pareto improvement, and we show
that this
is
WHLiile
not necessarily the case.^
Theorem
sheds doubt on
2 appears to give
for
Theorem
3 shows that competitive equi-
Pareto optimal when preferences are weakly separable (see also Theorem
which contracts with randomization are allowed). This
two reasons.
condition.
to the folk theorem, the rest of our analysis
general validity and applicabiUty.
its
libria are constrained
6 for the case in
some support
First,
Second,
it
most preferences used
in applied
establishes that the equilibrium
is
work
is
an important
satisfy this
resxilt
weak separability
constrained optimal relative to a very
strong notion in which the social planner has access to more instruments than the market (she
can monitor and specify expenditures
for all goods,
whereas contracts can only do so
for
goods
within a subset in the partition). This result suggests that, at least in most of environments
considered in apphed work, the inefficiencies emphasized by the folk theorem do not
Finally, in
Theorem
7
we show that when
mark environment with weak
there are only small deviations from this bench-
separability, competitive equihbria
(or are e-efficient for the right choice of e).
remain approximately
even
if
there
is
efficient
This result highlights another important concep-
tual point, that a large set of economically relevant environments
inefficiencies
arise.
may
not feature meaningful
a "generic" inefficiency result.
Overall, although our results do not imply that competitive equilibria are always efficient
in private information economies, they delineate a
equilibria have very strong optimality properties.
situations are a
good approximation
range of benchmark situations in which
They
also
show that when such benchmark
to the actual environment, efl&ciency will hold approx-
"Prescott and Townsend (1984a, b) emphasized the importance of contracts that allow for randomization for
efficiency in their analysis.
Such randomization
for the baseline efficiency results in
is
important
for existence in their
start with an environment that does not allow for such randomization.
when randomization
environment, but
is
not central
our or their framework. To highlight this and to simplify the exposition,
We
we
then establish the equivalent results
Theorem 5 generalizes Theorem 2, etc.).
we use for most of the analysis may be more relevant than this latter
weaker notion, since in a production economy the social planner can often achieve the allocations under full
monitoring by using taxes and subsidies. To simplify notation, we do not explicitly study such tax policies.
is
allowed
(e.g.,
'Nevertheless, the notion of efficiency
''See, for example, Atkeson and Lucas (1992), Golosov and Tsyvinski (2006), Golosov, Kocherlakota and
Tsyvinski (2003).
Suppose
imately.
example that the worker's preferences are defined at two
for
levels:
the
worker has preferences over a number of needs (such as food, entertainment, procrastination,
vacation, health care and so on)
and each one
of these needs
is
satisfied
by consumption of
various goods in the economy. Suppose also that there are only a few needs that interfere with
the worker's effort choice such as vacation, procrastination and health care. In particular, a
higher effort choice might
make
the worker enjoy vacations and procrastination less relative to
other needs (since she spends most of her time working) and enjoy health care more relative to
other needs (since the cost of effort decreases with level of health), but the worker's preferences
for other needs
consumption
may
not depend on effort
levels for the
Suppose
level.
also that
employers can monitor the
few needs that interfere with effort choice (but not necessarily the
consumption of the particular goods that make up these needs). This amounts
that firms can enforce
the
office,
applies
how
long a vacation the worker takes,
and how good a health care she
and shows that the equilibrium
stitutes a
receives.
will
how much time
to
assuming
she spends in
Under these assumptions, our Theorem
3
be constrained optimal. This scenario thus con-
counterexample to the conjectured suboptimality of competitive equilibria under
private information and
anonymous
trading,
and suggests considerable caution
in
appealing to
the above-mentioned folk theorem. Moreover, in this example, the folk theorem would suggest
taxing the goods that provide procrastination services (such as TVs), but our results indicate
wrong
that this might be the
tion of
TVs
recommendation since firms already monitor the consump-
policy
(and other procrastination goods) by making the workers come to the
'
work hours.
•-'
•
.
As the above
.
.
discussion clarifies, our paper
tionship of our paper to Prescott and
is
Townsend
.
during
office
.
-'
:
number
related to a
of literatures.
,
The
rela-
(1984a, 1984b) and to Greenwald and Stiglitz
(1986) has already been discussed. Another set of closely related papers are by Geanakoplos
and Polemarchakis (1986, 2008) and Citanna, Kajii and Villanacci (1998), which establish
the generic inefficiency of competitive equilibria in economies with (exogenously-given) incomplete markets.
Our work extends the Geanakoplos-Polemarchakis
endogenously incomplete markets (because of moral hazard).
environments constrained optimality may. result
itored.
It also
highlights that in such
the appropriate subsets of goods are
mon-
Similar issues arise in other economies with price externalities due to endogenously
incomplete markets.
for
if
with
results to environments
For example, Kehoe and Levine (1993) provide results similar to ours
an economy with participation constraints. Our paper
is
also related to
Citanna and
lanacci (2000), which establishes generic inefficiency of equihbria for a moral hazard
with exclusive contractual relationships
Our work shows
in
which the principal has
all
Vil-
economy
the bargaining power.
that their inefficiency result crucially depends on the assumption that the
4
principal has the bargaining power. Given the separability assumptions they impose on pref-
would be
erences, our results imply that the equilibrium
which the agent has
the "bargaining power,"
all
and the insurance market
is
i.e.
when insurance
competitive.
contracts are exclusive
t,v.';
,
,,
.,
which establishes the Pareto optimality of competitive equilibria
fully
separable
and Lisboa are worth emphasizing.
here, in
which
utility.
First,
A number
'
,
In addition to these works, the paper most closely related to our paper
with moral hazard and
opposite case in
efficient in the polar
'
Lisboa (2001),
is
in the context of
an economy
of key differences between our
work
Lisboa considers a special case of the model studied
utility functions are fully separable across all
goods and
effort
monitoring of consumption in any subset of goods. Second, Lisboa's analysis
order approach applying everywhere, which
is
restrictive
and not used
and there
relies
on the
is
no
first-
in our analysis. Third,
Lisboa's analysis contains neither our characterization results on inefficiency of comparative
equilibria nor our result
There
is
on approximate
also a large literature
efficiency.
on various
different aspects of
moral hazard
in general
equilibrium. Bennardo and Chiappori (2003) and Gottardi and Jerez (2007) discuss the prob-
lems that arise
in general equilibrium
transferability of utihty.
positive profits.
This
is
economies with moral hazard because of potential non-
They show how Bertrand competition might lead
an issue that also
arises in our
to equilibria
model and we provide
sufficient
with
con-
ditions (that are not very restrictive) for Bertrand competition to lead to zero profits. Bisin,
Geanakoplos, Gottardi, Minelli and Polemarchakis (2007) consider an alternative approach
to moral hazard in general equilibrium, where, in contrast to our setup, contracts are not
necessarily individualized. This introduces natural externalities across the actions of different
individuals signing the
Finally,
arise in the
some
same type
of the
same
issues
we emphasize
pubUc finance and mechanism design
(1987), Allen (1985), Atkeson
(2001),
of contract.^
Werning
and Lucas
in
literatures.
None
See, for example,
(1992), Guesnerie (1998), Cole
(2001), Kocherlakota (2004), Golosov
TowTisend (2006).
the context of general equihbrium also
and Tsyvinski
and Kocherlakota
(2006),
of these studies derive results similar to our
Hammond
and Doepke and
main theorems
in this
paper.
The
rest of the
paper
is
organized as follows. Section 2 presents the environment, defines a
competitive equilibrium, and establishes the existence of equilibrium. Section 3 introduces the
notion of constrained optimality and provides sufficient conditions under which the equilibrium
is
not constrained optimal. Section 4 introduces the notion of weak separability and presents
''Also related are recent
papers considering adverse selection
Gottardi (1999, 2006) and Jerez (2003).
in
general equilibrium, for example, Bisin and
our main result, which shows that the equilibrium
are weakly separable.
the preferences
Section 5 introduces the environment with stochastic contracts
which shows that the equilibrium
is
approximately constrained optimal when the preferences
are approximately weakly separable. Section 7 concludes.
from the main
text,
and
Section 6 presents our second main result,
generahzes the efficiency results to this setting.
results omitted
when
constrained optimal
is
Appendix A.l discusses additional
and Appendix A. 2 contains the proofs of
the results
all
stated in the text.
Environment and Equilibrium
2
Preferences
2.1
We
consider a static production
economy with a
set of (individual-specific) states of nature
denoted by S.
goods and
states,
continuum
of individual workers, denoted by
and use
|G|
the analysis and the exposition,
e|£;|}
..,
CR
by the function
when she
question
e
5
q,
we assume
The
a finite set.
whereby
such that
j/f
^
0,
M, with measure
that
all
and
normalized to
and a
finite
€ 5 to index
s
1.
There
G
q^ (e)
[0, 1]
is
\G\
W_^_
a
To simplify
workers have identical utility and identical
.
We
E=
represent this probability distribution
the probabihty of state
with a production vector
assume throughout that the
is
the worker induces a probability distribution
effort choice of
which ensures that each good g
across individuals.
G
use g E
~
exerts effort e (naturally with J^sesl^i^)
in turn, associated
is,
We
to denote the cardinality of these sets.
endowment (production) vector y £
over an
s
is
j^l
G
goods denoted by
In particular, each worker chooses an effort level e E E, where
production technology.
{ei,
and
finite set of
is
j/s
E W_^
'.
^
E
s
^°^ ^^' e
For each
in positive supply in
realization of states in
S
S
g,
for the
worker in
E E). Each state
there exists s E
some
(conditional on effort)
Thus, with a law of large numbers type argument there
states.
is
is
We
S
also
independent
no aggregate
uncertainty.^
We
assume that each worker has
VNM
preferences over consumption of goods and effort
choice represented by
U{x,e)
= ^qs{e)u{xs,e),
seS
where x^
=
(xl,...,x[
state utility function.
The
)
E
W^
is
the vector of consumption in state
Throughout, we use the notation Xs
=
s
and u () denotes the
{xs)\geG to designate vectors.
results generalize to multiple types straightforwardly provided that
worker type
is
observable and
contractible.
See, for example,
Bewley (1986)
or
Malinvaud (1973). Nevertheless, some care
right notion of integral in applying such a law of large numbers.
(1996),
is
sufficient here.
The
is
necessary in defining the
simplest approach, proposed by Uhlig
A=
=
{xi)\seS,geG to designate matrices. We refer to the pair (x, e) as an allocation, and let
V Id
X £^ denote the set of allocations. We make the following standard assumption
R|^
and X
CI
I
'
'
'
on the utiUty function:
,
•
.
,
,
Assumption Al (Preferences) The
state utihty function u(-)
entiable in Xs, strictly increasing in each if,
Throughout, the
effort choice of
.
,
the worker
is
and
strictly
is
twice continuously differ-
concave in
her private information, so there
hazard problem and employment contracts cannot be conditioned on
alized production vector
Motivated by the discussion
these reahzations.
Q = {Gi,
which
is
..,
publicly observable and
is
G\g\ } of the set of
equal to the set of
w^
expenditure
R+
£
all
goods
w = {w^riseSmeM
in the introduction,
on the monitoring subset Gm,
Gm
to denote the matrix
we consider a
for
each
m
£
M
=
p^
—
1.
denote the vector of prices by p G R\.
the corresponding price sub-vector, and by x'^
denote by p
use
where each element denotes the individual's expen-
For any subset of commodities G'
i.e.
We
as described below).
We
as the numeraire,
The
are traded in spot markets that operate after all
We
1
partition
{l,..,\Q\].
diture on each monitoring subset at a given state.
choose good
re-
contracts can specify the worker's
market clearing prices
at
The
effort choices.
employment contracts can condition on
The employment
goods, G).
production vectors are realized (and
a moral
is
a collection of disjoint subsets of goods the union of
(i.e.,
goods within each monitoring subset
Xg.
C
.
G, we
the corresponding consumption
sub-matrix.
2.2
A
Firms and Employment Contracts
large finite
set of firms
profits.
number
by J
=
Since, as
of firms can sign
{1, 2,
we
..,
|
Jj}.
employment contracts with the workers.
We
denote the
Firms are owned by the workers and maximize the expected
make
will see shortly, in equilibrium firms will
zero profits,
we do not
introduce additional notation to specify the allocation of their profits.
Throughout we impose
single firm.
An
exclusivity
and assume that each worker can only contract with a
employment contract between a firm and a worker
the worker's production,
y^, to
and prescribes consumption
that at this point
we
gives the property rights over
the firm and specifies worker's expenditures,
levels for goods,
x
=
{x^)\s^s,geG,
and
are not allowing stochastic contracts, which
probability distribution over outcomes.
w = {wT}seSmeM^
effort choice, e
E E. Note
would instead determine a
We return to stochastic contracts in Section 5.
c = {w,x,e), and we denote the set of contracts
We denote a contract by the tuple
C = rI^I^'I^^I X A (where recall that A = rI^'''''^' x E denotes the set of allocations).
that C is a subset of a finite dimensional space and its elements, denoted by c G C, are
by
Notice
simply
vectors.
An
incentive compatible contract
=
is c
C
£
(u;,a;,e)
such that the
effort choice
and
the level of consumption of goods are incentive compatible given prices p and wage schedule
>
w. More formally, this means that for a given market price vector p
G
(x, e)
0,
max U{x,e)
arg
(1)
{x,e)eA
We
— '^T
^sPs
subject to
effort level e, the
in (1) can
s
consumption choice
at each state s
denote the consumption choice at state
tion choice at
strict
all
states
is finite,
this
spondence e.[w,p).
>
and
c
2.3
G
C\p)
G C^{p).
is
solution.
e) ,e)
The
By
= maxt/ (x(zt>,p, e)
Berge's
upper hemicontinuous, and the indirect
recall that c
continuous in
its
(2)
solution
,
e)
Maximum Theorem,
utility function is
represented by a corre-
(3)
.
.
the correspondence e{w,p)
its
is
arguments (here
faces a
menu
less
the worker will never reject
this
^
'
of incentive compatible contracts, one from each firm,
reject all contract offers. In this case, she generates
and we adopt
is
continuous in
{c (i^ijOI-gj, and she chooses the contract that maximizes her
Vs,
arguments. The incentive
.
Worker's Contract Choice
each s (since she will be
the
simply a vector).
Each individual worker v
=
Maximum Theorem and
also define the indirect utility Junction
y(c,p)
each p
is
G argmax[7(x(ti;,p,
problem always has a
We
w'^ for each m.
then a solution to the following problem:
e
E
u{xs,e)
by the function x{w,p,e). By Berge's
effort choice is
to:
by the function Xj {ws,p,e), and the consump-
concavity of u(-) in x, the function x(-)
compatible
Since
s
uniquely determined as the solution
is
max
x^^pf •" <
subject to
Vs
and m.
be conceptually divided into two parts. Given an
Xs G arg
for
each
denote the set of incentive compatible contracts by C^{p).
The maximization problem
We
fo'^
utility.
The worker can
an alternative production vector
y^
<
also
ys in
productive without the firm). Since, as we will see, in equihbrium
all
contract offers, there
is
no
loss of
generahty in assuming that
assumption to simplify the notation. So when the worker rejects
,
contract offers, she solves
all
max U
subject to
Let
((
is
(x, e)
u;'"
=
XsP
<
(i.e.)
(4)
VsP for each
a;'^"p'^'")
,.
,x,e] as
outside option of the worker.
ifte
Hence, a strategy for the worker
index of the firm she chooses or
Ju
if
is
i^
& function J^:C
j
c
(:/,
|
=
p)
Rejecting every contract
p).
i-+
(p)'"^'
JU
{0} that specifies the
she chooses her outside option:
{c(i',j)}iej
earg_max Vic
'^•^-i
We
,;,':,
be a solution to the preceding problem, and define the contract
equivalent for the worker to accepting the contract c(j/,0
2.4
s.
{i^,j) ,p).
'
(5)
jeJu{0}
Firm's Problem
assume that firm
continuum of incentive compatible contracts, one
j offers a
worker, taking the price vector p, the contracts offered by other firms
and the worker
strategies
as given. ^'^
(J|y)t,£_/^
Lebesgue measurable function c(-,j)
Lebesgue measurable functions from
=
firm from an accepted contract c
[0, 1]
firm's profit
C^
[p).
(p),
Let
and
C ic'{py
let 7r(c,
given by:
{i(j,x, e),
K{{w,x,e),p)
Note that the
>
to
is
'^'\
-
each
^ j_r
is
x,
a
denote the set of
p) denote the profit of the
;
.
Z<
= Y^qs{e){ysp-
from a contract that
^j^
Formally, the contract offer of firm j
— C'
[0,1]
:
(c {i',j))
for
not accepted
is
equal to
0.
Hence, firm j
solves the following problem:
max
,
2.5
Tr{c{v,j),p)dv.
/^
,
,
,
,
(6)
Definition of Equilibrium
Let us refer to the economy described in the previous section with E. In this section, we define
a competitive equilibrium for economy £ and show that such an equilibrium exists.
Definition
1.
A competitive equilibrium
in
economy £
is
a collection of contract offers
[c{v,j)\^^j ^^j^ by the firms, a collection of strategies for the workers {^u)^^j>^, a price vector
p,
and accepted
contracts, [w
(i/)
,
x [v) e {v)]^^j^, such that
,
'°To justify the assumption that firms take the price of goods as given, we could put an exogenous limit
on the measure of contracts a firm can sign (a capacity constraint). If the capacity constraint is sufficiently
small and the number of firms is sufficiently large, each firm will be too "small" to influence equilibrium prices
while there will be sufficiently many firms to provide each worker with employment contracts. This extension is
straightforwad, but we present the model without capacity constraints to simplify the exposition.
,
1.
Workers' contract choice
2.
Firms maximize expected
problem
3.
is
optimal,
profits,
i.e.,
Goods markets
clear, i.e.,
for each j G J, the contract offer c{-,j) solves
x^^[v))dv
Indirect
facilitate the characterization of equilibrium,
>
qs (e)
>
for
and
if
p^
each s G
S and
e
also
>
0.
(7)
impose the following assumptions.
probability function qs
is
strictly positive,
G E.
Assumption A3 (Local Transferability) There
tions u^i ()
with equality
0,
Problem
we
Assumption A2 (Probability function) The
is,
the strategy J^ satisfies (5).
',eS
Firm Competition and the
that
J\f,
for each g E G,
y^ qs{e{v))[y^^ —
To
for each v G
(6).
M
2.6
i.e.,
exists a
monitoring subset Gi and func-
such that
w'^^'^i (•)
(8)
In addition,
u'-'^ (•)
satisfies:
lim
(9)
dxs
Assumption A2
first
is
Assumption A3
standard.
is less
part of the assumption, for example, will hold
if
standard, but relatively weak.
there
is
one good which
is
completely
separable from effort choice, and consumption of which can be monitored by the firm.
second part of Assumption
particular,
it
implies a
A3
is
The
The
a relatively weak form of the standard Inada condition (in
minimum consumption
requirement on the consumption vector, x^^
as opposed to separate requirements on the consumption of each good, xf ). This requirement,
along with Assumption A2, ensures that the worker's expenditure on the goods
positive at each state
can slightly increase
s, i.e.,
wl >
0.
The
first
(or decrease) the worker's expenditure
the incentive compatible effort level the same.
least
part of Assumption
A3
Without a condition that allows
drive profits to zero (see
for this
Gi
is
on the goods
in
Gi while keeping
Consequently, the assumption allows for at
Lemma
2 in
Appendix A. 2
wliile respecting
for a formalization).
type of utihty transfer, Bertrand competition
Bennardo and Chiappori, 2003). Since
10
strictly
ensures that the firm
a small amount of utihty transfer between the worker and the firm,
the incentive compatibility constraints (see
in
this
problem
is
may
not
already well
.
understood and
is
orthogonal to our main concerns, Assumption
questions of interest for us.
The
'
'
'
'
'
'.
,
'.
,
.
enables us to focus on the
c
.
-
-,-
:
following proposition provides an equivalent characterization for the equilibrium.
particular, a collection of accepted contracts
measure zero of them maximize the
part of an equilibrium
is
this paper, the proof of this Proposition
Proposition
1.
As with
(The Indirect Problem) Consider an economy E
Al-AS. Then, the prices and accepted
of an equilibrium
if
and only
The contract c{v)
(p,c{v)
contracts,
the other proofs in
=
that satisfies
Assumptions
['w{v) ,x{u) ,e{v)\^^jA, are part
a solution to
is
V(c,p)
'
(10)
•
ceC'ip)
.
all
all
.
but a measure zero of
i/
G
(H)
7r{c,p)>0,
subject to
for
but a
-
if:
max
-.
if all
Appendix A. 2.
in
is
and only
if
In
worker subject to the incentive compatibility
utility of the
constraint and a non-negative profit constraint for the firm.
1.
;.
A4
M
'
The goods markets
2.
clear
[cf.
Eq.
{7)j.
.
Moreover, at the solution to problem (10), the constraint (11) binds, that
makes
zero profits
m
each firm
equilibrium.
Existence of Equilibrium
2.7
The assumptions we made
so far do not guarantee the existence of an equihbrium.
S (p),
solution set to problem (10), denoted by
p,
is,
•
then the equilibrium
may
to be upper hemicontinuous
the constraint
the
not upper hemicontinuous in the price vector
In particular, the correspondence
not exist.
if
is
If
C'
set,
(p),
is
discontinuous.
S (p)
could
fail
In this setting,
the worker's incentive compatible effort choice can be discontinuous in the price vector p.
Nevertheless, because the elements of
effort choice, e), the
C^
correspondence C^
continuous even when effort choice
is
{p) are contracts, c
{p),
which
lies in
discontinuous in p.
to establish the continuity of the constraint set,
C^
Assumption A4
e
t
e
Rl'^l
(Effort Targeting) For each
=
{w, x, e) (and not simply the
the larger contract space C, could be
The
(p) (see
following assumption
Lemma
3 in
is
sufficient
Appendix A. 2).
e E, there exists a vector of utiUty transfers
such that
^qs{e)ts>^qsie)t,
ses
ses
11
for
each
ee E\{e}
.
(12)
•
This assumption
natural and quite weak.
is
It
loosely corresponds to the requirement
that firms should be able to induce (target) any level of effort
also not difficult to satisfy. For example,
it
holds
when
|£'|
<
they wish to do
if
It is
and the probability vectors,
\S\
each e E E, are linearly independent. In the commonly studied case of two effort
{Qs i^))ses ^°^
assumption holds when there are at least two states and the
levels, this
so.
Assumption A4 ensures the continuity of C^
to identical success probabilities.
S
turn implies that the solution correspondence to problem (10),
(by a version of Berge's
Maximum
do not lead
efforts
which in
(p),
upper hemicontinuous
(p), is
Theorem). The existence of equilibrium then follows from
standard arguments.
Theorem
(Existence of Equilibrium) Consider economy £
1.
A1-A4- There
Remark
1.
sufficiently strong
price vector p.
E.
A3 and A 4) An alternative to imposing Assumpcontinuous effort choice, e.g., E = [0, 1], and to make
to
assume
assumptions to ensure that the
The
effort choice
changes continuously in the
following set of sufficient conditions are typically adopted in general equi-
librium analyses of moral hazard economies with continuous effort
and
Stiglitz,
1991, 1993, or Lisboa, 2001):
between consumption and
effort choice,
c(-).
to high output
and low output).
concave in
these conditions,
it
e,
(2)
u
The
{xs, e)
(3)
The
state utihty function
=
v [xg)
— c (e)
states,
i.e.,
is
for
is
example, Arnott
is
fully separable
some function v {) and
S =
strictly increasing
can be shown that the first-order approach
is
(see, for
probability function q^ (e)
and the cost function c(e)
that the worker's effort choice
for
i.e.,
(1)
There are only two individual
a cost function
strictly
Assumptions
(The Role of Assumptions
A3 and A4 would be
tions
economy
exists a competitive equilibrium for the
that satisfies
{/i, /}
(corresponding
strictly increasing
is
and convex
valid,
which
in
e.
and
Under
in turn implies
continuous in p. However, these conditions are too restrictive
our purposes since they rule out nonseparable state
3 below), which play a crucial role in our
main
utility functions (defined in Definition
efficiency results in Sections 3
and
4.
Our
Assumptions A3 and A4 are considerably weaker than the standard assumptions, and as such,
they enable us to establish the existence of equilibrium for nonseparable as well as separable
state utility functions.
The
analytical difficulties that
emerge
state utility functions have been emphasized by Arnott
light.
Theorem
1
and
in the setting
with nonseparable
Stiglitz (1988b, 1993).
In this
could also be viewed as a methodological contribution to general equilibrium
analyses of moral hazard economies.
It is also
worth noting that allowing
for stochastic contracts as in Prescott
(1984a) does not bypass the need to impose Assumptions
A3 and A4
(see Section 5). Intuitively,
allowing for stochastic contracts convexifies the incentive compatibility
12
and Townsend
set,
C'
(p),
which sim-
.
plifies
the analysis in some dimensions. However, convexifying a discontinuous correspondence
does not necessarily make
continuous. Hence, the essential difficulty for establishing the
it
existence of equilibrium remains in the setting with stochastic contracts. This difficulty does
not arise in the Prescott and Townsend (1984a) economy, because contracts directly prescribe
consumption
levels for
each good and thus endogenous prices do not affect the set of incentive
compatibiUty contracts.
,
;
,
Generic Inefficiency of Equilibrium
3
This section formalizes the folk theorem for imperfect information economies, by providing
sufficient conditions
under which the equilibrium
inefficient.
is
We
start
by describing the
notion of efficiency used in our analysis.
An
allocation in our setting, {x, e) £ A,
effort-incentive compatible
is
optimal given the level of consumption prescribed, that
•
We
(f )
([2:
M, and
,
for each, e £
e {i')]^,^j^ is effort-incentive compatible
the resource constraints hold, that
V
/
Definition
timal
An economy-wide
2.
effort-incentive
is
if it
effort-incentive compatible
U {x {v)
,
qs{e{v)) {yl
U {x {1^) ,e {u})
>
e {v))
and
-
and
feasible
dv >
0, for
all
A-^
if
.
An economy- wide
{x [v)
xiiiy))
each
e {v))
€ A^ for each
is
M,
(13)
g.
constrained (Pareto) op-
and there does not
economy-wide allocation [x{v)
u £
,
alloca-
is,
compatible and feasible,
for
is
E
allocation \x{u) ^e{v)\.^^j^
feasible
the effort choice
is, if
denote the set of effort-incentive compatible allocations with
tion
v e
U {x,e) > U [x.e)
if
exist
,e{v)]^^^i^
another
such that
with strict inequality for a positive measure of
Consider an equilibrium of the economy E with price vector and accepted contracts,
(p,
\w {v)
,
X {v) e
,
wide allocation
Remcirk
2.
{'^)]^(zj>j)-
We
say that the equilibrium
[x (p) ,e{u)]^czj\f ^s
(Full
if
the
economy-
constrained optimal.
same informational constraints
(monitoring) technology.
notion of efficiency provides the
as the firms but with better contracting
In particular, the planner cannot observe the effort choice of the
worker, but can specify the consumption of
is
constrained optimal
Monitoring by the Social Planner) Our
social planner with the
of optimahty
is
arguably strong.
this notion helps us to develop our
all
goods
Nevertheless,
it
is
main point more
13
in the
employment
contract. This notion
natural for us for two reasons.
succinctly. For example, our
main
First,
result
(Theorem
a range of benchmark situations in which the equilibrium
3) delineates
this strong sense. Second, the social planner
is efficient
in
can approximate a constrained optimal outcome
according to our definition using more limited instruments,
i.e.,
a tax and transfer system.
For example, the government can reduce the consumption of a particular good by levying a
linear tax
on that good (though a tax and transfer system
Appendix A. 1 considers a weaker notion
of optimality
is
not equivalent to
full
monitoring:
and provides an example economy that
constrained suboptimal according to the strong optimality notion, but not according to this
is
weaker optimality notion).
The main
result in this section establishes sufficient conditions
under which the equilibrium
constrained suboptimal. These conditions are related to the notions of nonseparability
is
no
insurance, which
full
Definition
3.
we
Cm
define next.-"-
if
there exists a state
6
when
effort level is modified, that is
du
is
no
nonseparable (not
s,
a monitoring subset
Gm, <md two goods
utility
function
such that the marginal rate of substitution between g\ and §2 at state
91 ,92
There
is
Consider an allocation {x,e). The state
weakly separable) at (x,e)
full
and
(.3, e)
,'',;.
/dxf ^ du
insurance
changes
s
(X., e)
/^xf
eeE\{e}.
^"' '^"^
at {x,e), if there exists a
good g £
G
and two
such that the marginal rate of substitution for good g between states si,S2
G
S
not equal to
1,
states 5i,S2
is
that is
du{xs^^^e)Jdxi^
du{xs2,e) /dx%
The next theorem
our main inefficiency result.
is
'
'
shows that the equilibrium
It
strained suboptimal whenever the state utility function
'
is
nonseparable and there
is
is
con-
no
full
insurance at the equilibrium allocation for a positive measure of workers.
Theorem
2.
(Nonseparability and Inefficiency) Consider an economy £ that
Assumptions A1-A4- Let
[p,
[w
[v)
,x{v) ,e {v))^^j^\ denote the prices and accepted contracts
in an equilibrium. Suppose that there
M*
,
is
a positive measure set
the state utility function is nonseparable
allocation {x {u)
The
,
e (v))
.
and there
Then, the equilibrium
intuition for the result
is
satisfies
is
is
no
J\f*
full
C A^
such that for each v €
insurance at the equilibrium
not constrained optimal.
closely related to double deviations by the worker, that
is,
deviations in which a worker switches to a different effort level and reoptimizes her consumption
"As
Definition 4 below
preferences.
We
makes
it
clear,
"nonseparable" preferences are the converse of "weakly separable"
its easier to use than "not weakly separable" or
use the terminology "nonseparable" since
"weakly nonseparable".
-
14
new
of nonmonitored goods for the
there
no
is
A
the preferences are nonseparable
and
insurance at the equiUbrium allocation, double deviations bind in the incentive
full
compatibihty constraints. That
workers.
When
effort level.
social planner
they prevent firms from providing more insurance to the
is,
who can
also prescribe the
consumption of nonmonitored goods
not constrained by double deviations, and can therefore provide better insurance without
is
violating the incentive compatibility constraints.
worth noting that Theorem 2
It is also
utility functions are
open and dense
differently,
for
u
if
any £ G
utility functions, as defined in Defini-
continuous utility function (with the sup norm). Put
nonseparable, then eu
is
+
{1
—
e)
u
is
nonseparable
(0, 1).
The next example
Example
all
weakly separable and u
is
a generic inefficiency result, since nonseparable
weakly separable
(or
tion 4, are nowhere dense) in the set of
is
illustrates the intuition of
Suppose that there are two
1.
and low output, and two goods,
G=
Theorem
S =
states,
2.
The monitoring
{1,2}.
output
{h,l}, corresponding to high
partition
is
given by
G=
{G},
which implies that the firm can only specify wages and otherwise cannot monitor worker's
consumption. For simplicity, consider a partial equilibrium setting in which the relative price
of the goods
levels,
i.e.,
is
fixed
E=
firm
makes zero
that
is,
=
TT;
also that
is
equal to
,
=
2
0,
and
tt/j
=
relatively
is
y^p
work, and relatively
j^,
good
2,
0.
Assume
qk (e
is
=
1)
=
to leisure
Suppose there are two
effort
Assiune the
1/2 and
g/,
(e
than good
1.
=
0)
=
More
0.
Assume
specifically,
given by:
good
less
2 relatively
more
(in
when she works. Consider a
comparison to good
social planner
1, i.e.,
implements
she chooses x\
=
if
=
e
0.
=
1
(14)
1)
when she does not
social planner that chooses the allocation
x^) ,e), subject to effort-incentive compatibility
be seen that the
of only
1.^-
u{xs,e)=\n(^xl+(^^ + {l-eijxi^.
.,
f|,f
>
more complementary
Note that the worker enjoys good
((x^,
= p^ =
suppose p^
low output state and positive profits in the high output state,
profits in the
the worker's state utility function
.
1, i.e.,
{0,1}, which respectively correspond to shirking and working.
yip
good
and
and resource constraints.
It
can
and provides the worker with the consumption
Moreover, given that the worker does not consume
the state utiUty function in (14) imphes that there
is
no incentive problem. Thus, the
''The partial equilibrium setting is a special case of our model in which there is a linear production technology (that operates without moral hazard considerations) which can convert the two goods to each other. To
simplify the notation, we present our main results in an environment without this type of production technology.
Appendix B, which is available on request, shows that all of our results continue to hold when we introduce
a general (potentially nonlinear) production technology that is not subject to moral hazard. Also, again to
simplify the exposition, in this example we use functional forms that do not satisfy Assumptions A2 and A3.
15
That
social planner provides the worker with full insurance.
the
good
first
and the worker's
Next consider a firm that
full
enough
consume the bundle
(^) <
bundle. Since In
In
decision.
offers a
The
is
(15)
(i«
^My)-4y)-
wage contract
{wfijWi).
It
can be seen that the social
That
not incentive compatible.
(15), the
given the wages just
is,
worker would instead not work and consume a different
the worker can increase her utility with a double deviation
(5^),
which she changes her
= ^,
i]
= 5'"(t) +
insurance solution
planner's
=
by
utility is given
^(^.^)
in
the worker's consumption of
given by:
is
il
to
is,
effort choice
and reoptimizes her consumption
firm will instead offer the
wage contract [wh^wi) that
following equations:
_
'
is
new
for the
effort
the solution to the
.
(Budget constraint),
-
In {wh)
+
;t
In [wi]
>
In
The equilibrium wages and consumption
Wh =
The
Y^TTh, xl
equilibrium utility
is
Wh, x\
-Wh
(Incentive compatibility).
1
are given by:
=
0,and wi
'
=
given by:
//(.,e)
Comparing Eqs.
=
(
.
jo'^h,xj
=
=
wi, xf
0.
(17)
>
= iln(l..)+lln(l..)=ln(l..).
(17)-(18) with (15)-(16), note that "the firm
worker, and the contract offered by the firm
is
strictly
is
(18)
only partially insuring the
worse for the worker than the allocation
offered by the social planner.
Theorem
allocation.
hold.
in
A
2 establishes the inefficiency of equilibrium
natural question
To address
this question.
which any equilibrivim
is
is
under conditions on the equilibrium
whether- there are any economies for which these conditions
Theorem
8 in
Appendix A.l characterizes a
constrained optimal.
The
ensure that the equilibrium allocation of a worker always features
under
full
that there exists a shirking effort level which
is
any equilibrium. To
at
less
than
full
always preferred by
insurance, and which yields the firm (almost) zero profits.
16
economies
result essentially provides conditions
on the preferences and the technology such that Theorem 2 applies
we assume
class of
insurance,
tlie
worker
To ensure that every
equilibrium allocation satisfies the nonseparability property,
monitoring subset
between
Gm
and two goods gi,g2 £
Gm
and 52 always change monotonically
gi
Theorems
and
2
8 provide
some support
we assume
that there exists a
such that the marginal rate of substitution
in the effort level.
for the folk
theorem
for the inefficiency of
the
equilibrium. Recall, however, that these theorems rely on a strong notion of optimality which
essentially provides the social planner with a better monitoring technology
Remark
Theorems
2).
2
and
on whether a
8 are silent
than the firms
who
social planner
is
(cf.
also constrained
by the same monitoring technology can implement such a Pareto improvement. To address
Appendix A.l introduces a weaker notion of optimality which constrains the
this issue,
planner with the same monitoring technology as the firms.
example economy with nonseparable
Theorem
8,
but
is
utility,
which
is
This appendix also provides an
constrained suboptimal as implied by
The example shows
weakly constrained optimal.
social
that the inefficiency of
equilibrium established in Theorems 2 and 8 in part stems from the strong notion of optimality
wliich gives the social planner a technological advantage in monitoring. This suggests that care
must be taken
in invoking these theorems.
Efficiency of Equilibrium under
4
We
next provide our main result, which
result
shows that the equihbrium
separable.
The next
Definition
subset
Gm
The
4.
o,nd
is
the converse of
constrained optimal
state utility function u(-)
level.
du{xs,e) /dxf
That
Gm,
Note that the state
(x, e)
is
Separability
Theorem
when worker
weak
In particular, the
2.
preferences are weakly
separability.
weakly separable
if,
for any monitoring
ihs marginal rate of substitution between gi
and 32
is
is,
du{xs,e) /dxf
du[xs,e)/dxf = du{x,^e)/dxf
any allocation
is
Weak
definition formalizes the notioir of
two goods gi,g2 G
independent of effort
at
-
/
utility function
^"' '"''' ''^''~ "
^^
weakly separable
is
if
\g\
"^ ^
^^
and only
,
~
'^"' ^'^
if it is
G A. Our next result shows that weak separability
is
^ rp
" ^^
nn\
^^'^
not nonseparable
sufficient for the
competitive equilibrium to be constrained Pareto optimal.
Theorem
3.
(Efficiency under
Assumptions A1-A4- Assume
for the
Separability) Consider an economy £ that
satisfies
also that the state utility function u(-) is weakly separable.
any equilibrium of the economy £
The
Weak
intuition of this result
is
is
Then,
constrained optimal.
that,
under weak separability, the social planner chooses
worker the consumption bimdle which the worker would have chosen by herself
17
in
the anonymous trading market.
competition
A
among
Hence, there
no benefit to additional monitoring, and
is
firms leads to the allocations that the social planner would have chosen.
complementary intuition
that double deviations in which the worker changes effort level
is
and reoptimizes her consumption bundle accordingly are not valuable. This further implies
economy does not change the
that the relative price changes caused by other contracts in the
We
insurance-incentive trade-off for a worker, rendering pecuniary externalities ineffective.
demonstrate this theorem with a simple example
Example
in
Consider the same setup as in Example
2.
which preferences are
1
with two differences.
that price externalities do not create inefficiencies in this setting,
than
relative prices
Example
in
that
1,
we consider the
is,
fully separable.
emphasize
First, to
we consider more general
=
price vector (p^
l,p^ €
R-).)
(we
continue to assume that the prices are fixed). Second, we assume that the worker's state utility
function
is
given by
=
u{xs,e)
where u^^^
V
(v, e) is
and decreasing
and
strictly
V [xl,xl)
=
and
1
a scalar valued function that
in
and v (x^jX^)
e,
is
is
,
strictly increasing
{
{x\)
as the
2.
of the need, v,
The
+
=
*"
(ig)
^
j
consumption
>
e
0).
{v, e)
is
u^^P {v, e)
concave in
strictly increasing
=
—
In [v)
which
is
satisfied
utility function, u(-), is
,
/dxl
e)
and
by consuming
separable, since
du [xs,
e
gives the worker's utility corresponding the level
Note that the state
effort choice, e.
is
strictly
Here, the inner function v {x\,x1^ can
level of a particular "need,"
outer function u*^^
and the
some
for
and
a scalar valued fimction that
concave in both arguments (a particular instance
be thought of
goods
u"'P{v{xl,xl),e)
^
weakly
.
,
dv {xl,xl) /dx\
'
du{xs,e)ldxl
which
is
independent of the
Consider
first
((x^, Xj^,x^,a:p)
,
e)
a
"
who determines an
each worker.
on
this assumption).
The planner maximizes
effort-incentive compatibility
effort-incentive
compatible
allocation
Suppose, for simplicity, that the planner offers every
worker the same employment allocation (the result
rely
'
effort choice.
planner
for
"
dv{xl,x1) /dxl'
and resource
in
Theorem
3
is
more general and does not
the representative worker's utility subject to
constraints, that
is:
max qH{e)u''P{v{xlxl),e) + {l-qH{e))u^'P{v{x],xf),e)
{x,e)SA
subject to
'^qs{e){xl+p^x1)
seS
solve this problem, the planner
^qs{e)TTs,
seS
U {x, e)>U (x, e)
To
=
for all e
£
:_
£^.
computes the value from implementing any
18
e
G E.
The
planner then implements the effort level that yields the highest utiUty to the worker. Similar
to the analysis in
of the
weak
equivalent to
Grossman and Hart
(1983), the social planner's
how much
deciding
simplified in
is
view
In particular, the planner's problem
separability of the state utility function.
first
problem
of the need to provide in each state, Vs,
is
and then deciding
the optimal consumption bundle that provides this level of the need. Hence, the optimal utihty
from implementing
£
e
E
given by
is
max
qh{e)u''P{vh,e)
s€S
^"''
(^'^'
E
^
^~)
seS
each
s
£
—
(i^,a;^)
that, given Vs, the planner
for
would
{/i, I],
v^
and v
(^xj,xf)
the vector x^
is
G E,
subject to
V
=
vi.
= min
=
(^xl,x'l)
this level of
the solution to:
xseK
is
^°' all e
(^'^ ^)
minimize the cost of providing
like to
Wpi {vs)
This
^"^
9^ (^)
s€S
and two additional constraints v
is,
(20)
ses
Y. 1^ (^)
the need. That
{l-qh{e))u'''P{vi,e)
Y^qs{e){xl+p'^xl) =^qsie)TTs
subject to
Note
+
1
x]
xs
+ pJ2-2
,
|G|
Vs.
a strictly convex minimization problem that provides a one-to-one relationship between
Wpi () and the
optimum
choice of the consumption vector Xg-
By
duality, the
optimum vector
Xs also maximizes v (x^,^^) subject to the expenditure being not greater than Wpi
(u^).
Using
these observations, problem (20) can be rewritten as
max
subject to
E
^
9^
""'
'^^^
(^~)
,
r
qs (e) Wpi {vs)
^
=
?s (e)
(^ (^5,5^)
,
e)
(21)
t^
seS
y^orrcrfls (e)
^.cc9s
(e)
I
I
maxf
maxr
Next consider the problem of a firm
igi
„igi
,
,
o
offering a
19
,
,
o
o-o
,
,i
,-
.^
u^^^ (v (xl.x'i) ,e)
w^^^ (u (ij, z?l
,
e)
|
for
]
>
each e £ £.
wage contract {Wg,Wb}- To implement
effort
level e
£ E, the firm will solve
^
max
^
subject to
max
qs (e)
Qs (e)
Eses<ls
=
it;^
^
{xlx^)
u^'^p {v
e)
(22)
1
g^ (e) tt^
(e) (^max|.^^j^K;i
^'^^ (^
(^l.^') ,e) j
ji+p2j2<,,^}
,
A
,
comparison of problems (21) and (22) shows that they are equivalent.
social planner
and the firm
will receive the
same
utility in
and a firm that can only
who can monitor
The
weak
will
is
(u^, u^,
its first
and the worker
eff'ort level, e,
is
not valuable,
..,
Suppose that the state utihty function can be written as
=
u'^l, e)
u^^P [v'
is
{x^^)y
(:r?=)
,..,
\M\ arguments, and for each m, v"^
its
:
in (8) are
W_^.
number
(23)
'
,
strictly increasing
I
i—>
M
is
arguments. This functional form
distinct set of goods.
Theorem
is
\G '"'
and
strictly
a scalar valued function
is
intuitive, as
it
implies
of higher order needs (such as food, entertain-
ment, procrastination, vacation, health care, education
consumption of a
J^^l (xf'^') ,e)
a scalar valued function which
that the worker receives utility over a
represented as
£;.
also suggests a general class of preferences that satisfies the
increasing and concave in
separable. Thus,
e
Consequently, the
each case. Hence, in this example, monitoring
example
u{x,,e)
which
e
a wage contract provides incentives as well as a social planner
ofi'er
separability condition.
concave in
each
worker's consumption.
analysis in this
where u^^^
choose to implement the same
for
etc.)
each of which
is
provided by the
Clearly, the state utility function in (23)
is
weakly
3 implies that equihbria of economies in which preferences can be
constrained optimal.
This analysis the suggests that,
monitor the consumption of broad aggregates that interfere with
effort choice,
if
firms can
then the equi-
librium will be efficient even though firms are unable to monitor workers' exact consumption
•"
choices.
5
Equilibrium and EfRciency with Stochastic Contracts
This section extends our setup to allow
tic contracts."
Allowing
nonconvexity of problem
for
for contracts
with random outcomes, briefiy "stochas-
randomization can be beneficial in this economy because of the
(2) in Section 2 (e.g., Prescott
20
and Townsend, 1984a, 1984b, Arnott
and
analysis in this section shows that allowing for stochastic contracts
Our
Stiglitz, 1988a).
does not change the efficiency properties of this economy. In particular, analogues of our main
Theorems
results,
Given a
V (Z)
and
2
apply for the economy with stochastic contracts.
3,
dimensional space Z, we
finite
B
let
{Z) denote the Borel cr-algebra of
denote the set of probability measures over (Z,
B (Z)). We endow P
t]
tj
(
by
U^
(77),
measm-e,
contract
and
j.l
/i,
equal to
it is
/,
U {x, e) drj.
^^^^
iC =
£ V{C), over the contract space
we
V {A)
S
let M|(i,e)
Similarly, a stochastic contract
x
Kl,.
is
is
resolved after the effort decision
and then learns which contract she
effort level
contract uncertainty
A,B[C^\. Given a
stochastic
is
\i
can be
With ex-post randomization,
made, that
is
is,
the worker chooses
With ex-ante randomization, the
will receive.
resolved before the effort decision
is
a probability
resolved, a stochastic contract
interpreted as having either ex-ante or ex-post randomization.
the contract uncertainty
is
denote the marginal measure over the allocation space A.
Depending on when the contract uncertainty
an
(Z) with the weak*
A stochastic allocation in our setting is a probability measure over the allocation
x E,B{A)]. The worker's utihty from a stochastic allocation
A = M.\.
is denoted
topology.
space
Z and
made,
the worker learns her
i.e.,
contract before she chooses an effort level (see, Arnott and Stiglitz, 1988a, for further discussion
and the
role of
For analytical and notational convenience, we
each type of randomization).
analyze the case with only ex-ante randomization. All of the results in this section generaUze
to the case with both ex-ante
and ex-post randomization.^'^
Given the assumption of only ex-ante randomization, a stochastic contract
compatible
tracts, that
if
and only
is,
supp[ji)
a probability measure
support
if its
C C
fi
£
V
{p).
lies in
Equivalently, an incentive compatible stochastic contract
profits of the firm are respectively given
by Jqii„)
deterministic case, each worker
menu
faces a
and chooses the contract that maximizes her
tracts to
maximize
offers as given.
its
We
expected
incentive
the set of deterministic incentive compatible con-
[C^ {?)) The expected
i'
11 is
profit,
utility of the
V {c,p) d/x
and
worker and the expected
/(^//
\
tt
(c,p) dfi.
of incentive compatible contracts
utility.
is
<
As
in the
tJ-iuj)
\
i
Each firm j offers a continuum of con-
taking the workers' strategies and other firms' contract
denote the economy with stochastic contracts with £^, and we define the
equilibrium for this economy as follows.
Definition
5.
A competitive equilibrium
ible contract offers
<
i^-t^
'
It
is
well
known
economy £^
is
a collection of incentive compat-
by the firms, a collection of strategies for workers, [J^j
j) \
a price vector p, allocations
in
[/^yji^gyv"
'
•s^cft
that:
workers' contract choice
is
optimal, films
^,
max-
that ex-post randomization in this setting could be useful to provide the worker with
additional incentives (see Arnott and Stiglitz, 1988a).
However, as noted by Bennardo and Chiappori (2003),
this additional incentive provision does not interfere with the existence or efficiency properties of equilibrium.
21
imize expected profits, and goods markets clear, that
/
- xl)dii^dv>Q,
with equality for p^
>
Q.
(24)
J(w,x,e)eC'(p)'^g
Jj\f
The
"S^ qs{e){yl
/
for each g:
is,
following result
is
the counterpart of Proposition
1
and Theorem
1
with stochastic
contracts.
Theorem
omy £^
(Existence of Equilibrium with Stochastic Contracts) Consider an econ-
4.
Assumptions Al-AJ^.
that satisfies
economy £^. The prices and accepted
only
1.
There exists a competitive equilibrium for the
contracts, (p,
P'^''^^
°f
'^'^
equilibrium if
and
if:
For
all
M
hut a measure zero of workers v G
max
subject to
the contract
,
/
V(c,p)dLi
/
TT
The goods markets clear
/i^
is
a solution to
(25)
{c,p)dfj,>0.
Jc'(p)
2.
'^^^
[Mi/]i>eA/')'
(26)
.
Eq. (24)/.
[cf.
Moreover, at the solution to Problem (25), the profit constraint (26) binds, that
firm makes zero profits
m
equilibrium.
..
every
is,
•,
Analysis of the efficiency of the equilibrium closely parallels the case with deterministic
contracts. First,
we provide the analogous
definition of constrained optimality with stochastic
contracts.
Given the assumption of only ex-ante randomization, a stochastic allocation
incentive compatible
if
and only
compatible allocations, that
effort-incentive compatible
is,
and
if
the support of
r;
£
P
{^^)-
feasible
if
rj
is
effort-
the set of deterministic effort-incentive
lies in
^n economy-wide
and only
rj
if 77^
6
P
stochastic allocation
{/^^) for
\'n,j\^^j^ is
each u and the resource
constraints hold:
'
Tqs{e)(yi-xl)dr^,dv>{}ioxe^chg.
/ /
(27)
JAfJ{x,e)eA'^
Definition
is
An
6.
economy-wide stochastic allocation
effort-incentive compatible
and
feasible,
and
compatible and feasible economy-wide allocation
u £
N with
strict inequality for a positive
\ri^]j^^j^
is
constrained optimal
if it
there does not exist another effort-incentive
[fli,]^i^j^
measure of
such that
U^
(t)^)
> U^
(7?^)
for
all
v.
Consider an equilibrium of the economy 8^ with the prices and accepted contracts,
{pAi^u]u€J\f)-
cation
^^
\^i'i,\(x,e)\
^^-y ^^^^ *^^
(iKf
^^
equilibrium
is
constrained optimal
constrained optimal.
22
if the
economy-wide
allo-
,
Our next
result
Theorem
the analogue of
is
sufBcient conditions under which the equilibrium
we generahze
Definition
0.
The
7.
Consider a stochastic allocation
state utility function u(-)
is
between g\ and 52 o^ state
nonseparable
'^
no
full
The
Theorem
5.
is
:"
is
if there
{A*)
>
exists a state s,
a
such that
t]
such that the marginal rate of substitution
level is modified, that
A
-
and
G
good g E
-
.
E\{e}
and two
/
-
.
e
e
is,
,
.
states Si,S2
6
S
.
our main inefficiency result for the economy with stochastic contracts.
(Inefficiency
under Stochastic Contracts) Consider an economy E^
equilibrium. Suppose that there
there exists a
A
A* C
set
at {ri,A*),
at {r],A*) if there exists a
Assumptions A1-A4- Let
satisfies
insurance to stochastic allocations.
for each [x,e) e
.
•
following
du{x.,e)ldxf
insurance
such that
Gm
changes when the effort
s
du{x.,e)/dxf
is
full
and a compact
77
and establishes
not constrained optimal. To state the result,
is
the notions of nonseparability and no
monitoring subset Gm, ond two goods 51,52 £
There
2 for stochastic contracts,
compact
A* C
set
nonseparable and there
is
is
no
(p, [/^t/]^gj\/)
denote the prices and accepted contracts in an
a positive measure set Af*
A
with
tJ-i/\{x,e)
(^*)
insurance at
full
that
>
C Af
such that for each v £
such that the state
(Miy|(a;,e)i
^*)-
utility
J\f*
function
Then, the equilibrium
is
not
constrained optimal.
We
next provide the analogue of
shows that, when the
contracts
is
Theorem
utility function is
3 for stochastic contracts.
The
following result
weakly separable, any equilibrium with stochastic
constrained optimal.
6.
(Efficiency under Stochastic Contracts) Consider economy £^ that
Assumptions AI-A4. Assume
any equilibrium of
6
Theorem
the
also that the state utility function
economy £^
is
u ()
is
weakly separable. Then,
constrained optimal.
Approximate EfRciency of Equilibrium
As noted
in Section 3,
weak
separability
is
a nongeneric property (in the sense that any weakly
separable utility function will become nonseparable after a small perturbation).
there
satisfies
is
also a sense in
which such genericity results
economic importance of certain types of
may
However,
not be relevant for understanding the
inefficiencies. In particular,
such genericity results do
not preclude the possibihty that in most economically relevant situations equilibria might be
23
"approximately" efScient. In the present context, weakly separable utility functions might be
a fairly good approximation to most economic situations (including almost
in applied work).
when
Thus, what might be relevant
there are only small deviations from such
We
this question.
its
Definition
weak
A/",
we
investigate
an alternative economy with weakly separable
"close to"
is
8.
For each
e
>
0,
an economy-wide allocation
utility
[r]^]i,^j^ is
e -constrained
optimal
compatible and feasible, and there does not exist another effort-incentive
compatible and feasible economy-wide allocation
V S
separability. In this section,
inefficiencies
equilibrium will be approximately constrained optimal.
if it is effort-incentive
all
whether there are significant
introduce a notion of approximate efficiency (e-constrained optimality) and
show that when the economy
functions,
is
cases considered
all
[f)u]^^j^
U^
such that
with strict inequality for a positive measure of v G
> U^
(7)^,)
{r]^,)
+
e
for
J\f
Consider an equilibrimn of the economy £^ with the prices and accepted contracts,
^^
(Pi [Mi/li/eA/')-
^^-y ^^^^ ^^^
^
location [fi^lr^g^^
The notion
optimality
(cf.
equilibrium
is e- constrained
is
e-constrained optimal
optimal.
of e-constrained optimality
is
economy-wide
if the
al-
.
a generalization of the notion of constrained
Definition 2) since the two notions are equivalent for e
=
0.
We
next introduce
the notion of a perturbation of the state utility function. Recall that, by Assumption A3, the
'
state utility function has the representation
u{xs,e)=u^'
for
some functions
u'^^ (•)
and
',
(xfi)
+u^\^i (a;f^^^e)
For simplicity, we keep the transferable component,
u'^^'^i (•).
u^^ (), unchanged and we focus on perturbations of
generahty). Let
u*-''
"
I
(
u^\^^
Note that any state
U
-u'-'^^i
() (this
is
without loss of any
() denote a continuously differentiable, strictly increasing, strictly concave
and bounded function, and consider the
^^
,
:
R^
^^'
X
set of state utihty functions:
u
E
utility function
with the sup norm, which makes
U
u G
U
For a state utility function u G
satisfies
into a
ZY,
satisfies
we
.,
Assumption Al and
Assumptions Al and A3.
normed vector
space,
'
is
,
bounded.
We endow the
set
^"^
and thus a metric space.
B {u,S) = {u & U \\u — u\\ < 5} denote the
functions in B (u, 5) can be thought of as small
let
\
5 neighborhood of u.
When
d
>
0,
the
The assumption that u is bounded is without loss of generahty, because each worker's production of each
good g is bounded above by maxsgsj/f, which implies that the utility function can be restricted to a bounded
consumption set. More specifically, for any unbounded state utility function, u, there exists a bounded state
utility function, u, which agrees with u on the relevant consumption set and which leads to the same equilibrium
set.
24
perturbations of the function u.
B (u, 5)
with
v.
is
weakly separable, then the state
are close to being weakly separable.
utility functions in
Theorem
satisfy
If
7.
B (u, 5)
Our next
are approximately efficient (e-constrained optimal).
(Approximate Efficiency)
Assumptions A1-A4 and that
B (u, 6),
We
shows that equilibria of economies
result
^ denote
Let \S^ (w)]
the class of economies that
differ only in the state utility function,
>
a weakly separable state utility function u £ U. For any e
u E
utility functions in
then any equilibrium of the economy £
0, there exists 6
u e U. Consider
>
such
that, if
{u) is e-constrained optimal.
provide a sketch proof of this result, which
completed
is
present a social planner's problem for this economy, which
We
Appendix A. 2.
in
first
useful to characterize the efficiency
is
properties of the equilibrium. Consider the problem of maximizing an (equal) weighted utihty
and resource constraints:
of the workers subject to effort-incentive compatibility
max
subject to
/
/
U {x,e)dfjjjdu
/
/
\^ qs
Jn J{x,e)eA' f^
Note that problem
(28)
is
—
(e) (yf
a linear optimization problem
(28)
xl)dr]^,dv
>
0.
and the constraint
set,
V
{A^),
is
convex. Hence, problem (28) has the same optimal value as the following simpler problem:
max
U{x,e)di)
/
veV{A')
subject to
'S2qs{e){ys -^i)d'fi>0,
/
J{x,e}eA'
where
t)
e
V
(A^)
is
(29)
J{x,e)eA!
^g5
the average measure defined by:
V (a)
=
f
r,^
(a) dv
each
for
i
£
S
[A')
(30)
.
In other words, the social planner can be thought of as choosing a single stochastic allocation
to
maximize a worker's expected utihty subject
to resource constraints. Let J7p;^„„g^ [u) denote
the optimal value of problem (29) for the economy with the state utihty function u G U.
The next lemma
characterizes approximate constraint optimality of the equilibrium by
comparing the optimal value of the
social planner's
equilibrium problem (25). To state the result,
when the
Lemma
price vector
1.
is
let
problem
U^^
is
") denote the value of
given by p and the state utility function
E^
Consider an equilibrium of the economy
the equilibrium
(p,
(29) with the optimal value of the
e-constrained optimal
In particular, the equilibrium
is
if
and only
if
constrained optimal
25
is
u £ U.
with the price vector p. For any £
U5^^^^^ (u) G [U^
if
problem (25)
and only
if
{p,
u)
US^^^^^ {u)
,
U^^
=
(p,
U^^
u)
>
0,
+ e].
[p, u).
2
,
Given
this
lemma, Theorem 7
problems (25) and
erties of
intuitively follows
To
(29).
equilibrium of the economy £ iu)
constrained optimal.
is
=
establishes that a version of Berge's
^Sanner (^)
^^
it
<p e M+
upper hemicontinuous
U^ (p, u)
(31)
problem
applies to
Similarly, Ug„ (p, u)
in u.
I
p
is
and thus
(29),
a continuous function of
is
an equilibrium price vector of £
These observations, along with Eq.
(u)
this
(32)
>
imply that U^^^^^^, (u)
(31),
are close to each other for a state utility function, u, that
Appendix A. 2 formalizes and completes
of u.
this implies
can be seen that the equilibrium price correspondence,
P (u) =
and
1,
imphes that any
Next, the analysis in Appendix A.
(u).
Maximum Theorem
^ continuous function of u.
{p,u). Moreover,
By Lemma
6
U^,{p,u),
any equilibrium price vector p of the economy E
for
Theorem
see this, first note that
UXnner{'a)
is
from Theorem 6 and the continuity prop-
lies in
a neighborhood
argument, establishing a proof of Theorem
7.
Theorem
useful.
5
>
0.
7 also highlights
Take the
set of
why
the notion of generic inefficiency
economies corresponding to
With the same argument
dense within the set B{u,5).
3.
Theorem
that
utility functions in
the set
B {u, 5)
But our
shows that
result
all
for
some
nowhere
as above, weakly separable utility functions are
have equihbria that are approximately
Remark
not always economically
is
of the corresponding economies
efficient.
(The Role of Stochastic Contracts in Approximate Efficiency) Note
7 concerns the
economy £^ with stochastic
analogous approximate efficiency result
contracts,
and we do not have an
an economy £ with deterministic contracts.
for
conjecture that the result generalizes to economy £, but this conjecture
is
We
not straightforward
to prove.
The proof
to be
Theorem
of
no analogue of
set of incentive
7 does not generalize to
Lemma
1 in
C
sarily characterized as the solution to a
in
£,
mainly because there appears
that setting. In particular, due to the nonconvexity of the
compatible contracts,
example economies
economy
(p),
the Pareto frontier of
economy £
weighted social planner's problem.
which the equilibrium
is
is
not neces-
One can
construct
constrained optimal; but neither problem (28)
nor problem (29) (nor any other weighted social planner's problem) yields the worker the same
utihty as the equilibrium. Since our proof of
Theorem
7 exploits the continuity properties of the
social planner's problem, this proof does not generalize to
economy
£. Stochastic contracts are
useful in this context because they convexity the set of incentive compatible contracts, which
26
in turn enables us to characterize constrained optimality using a social planner's
Lemma
formalized by
problem (as
1).
Conclusion
7
This paper investigated the efficiency of competitive equilibria in environments with private information. Prescott and
Townsend (1984a, 1984b)
petitive equilibrium in such environments
can
fully specify
to situations in
when
establish the constrained optimality of
(insurance,
Though important,
consumption bundles.
which individuals are allowed to trade
anonymous trachng
employment
in
"folk
or credit) contracts
these results are not apphcable
anonymous markets.
to be an essential feature of competitive equilibria. Less
structure and efficiency of competitive equihbria in the presence of such
A
theorem" originating
work of
in the
Stiglitz
com-
is
We
view such
known about the
anonymous
trading.
and coauthors maintains that compet-
equihbria are always or "genericaUy" inefficient in such environments. This folk theorem
itive
has widespread appUcability in both applied models and in policy discussions, though
been formally investigated. This paper
of a general equihbrium
to workers
To
economy with moral hazard.
who choose an
effort level that is private
endowment and production
distribution of
equilibriimi
critically reevaluates this folk
and characterize some of
its
vectors.
theorem
within a subset.
We
how
context
information and affects the probability
We establish the existence
of a competitive
properties.
employment contracts can
the partition but camiot regulate
in the
has not
In our economy, fo-ms offer contracts
investigate the efficiency properties of competitive equilibrium,
itoring partition such that
it
this expenditm^e
we introduce a mon-
specify expenditures over subsets in
is
subdivided among the commodities
say that preferences are nonseparable (or not weakly separable)
when the
marginal rate of substitution across commodities within a subset in the partition depends on
the effort
level.
We
prove that the equihbrium
rium allocation involves
result
main
is
less
than
full
is
always inefficient when a competitive equilib-
insurance and preferences are nonseparable. While this
consistent with the folk theorem on the inefficiency of competitive equilibrium, our
result
shows why such inefficiency does not always
monitoring.
We
by preference
is
show that when there
is
weak
arise
partial
separability in preferences, a condition satisfied
used in most applied theory work, competitive equilibria with moral hazard
are constrained optimal, in the sense that a social planner
consumption
and can be mitigated by
levels
who can
regulate and monitor
cannot improve over these competitive allocations.
We
also
all
show that
equilibria in economies that have utility functions that are approximately weakly separable
will
be approximately
efficient.
These
results
imply that the strong suboptimality claims of
27
the folk theorem for competitive equiUbria in private information economies
exaggerated. At the very least, considerable care
equilibria are inefficient
is
may be somewhat
necessary in concluding that competitive
and government intervention
is
necessary without knowing the details
of preference and information structure.
Our
results also
on the monitoring
emphasize that the efficiency properties of competitive equihbria depend
partition,
which
raises the question of
how
the monitoring partition
termined in practice. In related work, we develop a framework
equilibria in
competitive
other things, this framework shows that endogenous monitoring will create
another force towards
is
bounded above by the
partition (which depends on the worker's preferences).
cost of monitoring this partition
cost of monitoring a particular
This further implies that, when the
sufficiently small, the competitive equilibrium is approxi-
is
mately constrained optimal, despite the costs of monitoring that
who does not
equilibrium compared
efficiency. In particular, additional welfare loss in
to the constrained efficient allocation
planner
de-
which firms pay a cost to choose which subsets of commodities and actions to
Among
monitor.
for the analysis of
is
it
incurs (relative to the social
incur them). This result reinforces our point that considerable care
is
necessary in invoking the folk theorem about the inefficiency of competitive equilibria with
private information.
A
Appendices
A.l
Omitted Results.
The appendix
:
presents results omitted from the
\
main
Class of econoinies in which any equilibrium
text.
is
inefficient.
Theorem
text estabhshed sufficient conditions under which the equihbrium allocation
suboptimal.
The next
the conditions of
result identifies a class of
Theorem
features less than
full
2,
and
insurance,
thus,
profits.
To ensure that every
is inefficient.
we assume
always preferred by the worker under
economies
full
in
To ensure
insurance,
same monitoring
and which
[£ (^)]gg(o
I)
which
differ only in the
satisfies
yields the firm (almost) zero
The
we assume
that
result then follows
.
For Inefficiency) Consider
a class of economies
parameter 6 defined below. Suppose that
28
is
partition such that the marginal rate of
.-
(Sufficient Conditions
constrained
that there exists a shirking effort level which
from Theorem 2 (proof omitted).
8.
main
that the equilibrium always
substitution between the goods change monotonically in the effort level.
Theorem
is
which any equihbrium
equilibriurri has the nonseparability property,
there exists two goods within the
2 in the
1.
There exists an
effort level, Cshirk
E
x £
\ {eshirk}
and
qsi„„ {eshirk)
=
(x
o,nd
0,e
=
=
Cshirk)
^
'is
]RI^|?^
—
S
There also exists a state sio^ £
.
Moreover,
0-
> u{x,e)
£ E, such that u{x,eshirk)
U {Q,e shirk) <
maxegjr
for
such that
i7 (y, e),
=
ysi^^
the allocation
i.e.,
than the allocation that offers no insurance (and
isss desirable
6
e
all
lets
the worker choose the effort level).
There exists a monitoring subset
2.
strictly increasing in e
ECR
E
There exists 6 £ (0,1) such
least
that,
Gm
o-i^d
Gm
two goods gi,g2 S
for any x £
9,
is
yg^gi
'.
Ri,_Jjf'
<
for each 9
such that a h
Assumptions AI-A4
one competitive equilibrium, and any equilibrium of the economy 8
hold,
[9) is
£
{9)
has at
not constrained
optimal.
Alternative notions of efficiency.
We
next consider a weaker notion of optimality than
studied in the main text. This notion of optimahty constrains the social planner with the
same
monitoring technology as the firms.
market
C
price
and contract allocation
feasible
if the
contracts are incentive compatible given the price vector,
9.
for each u £
{p)
strained optimal
=
(p, [c{v)
{w
{v)
N
if it is
,
and
[w{v)
feasible
{v)]^,^j^),
and there does not
such that
U {x {u)
exist
,e{y))
,x{i') ,e[v)\^^j^^,
The pair
is
i.e.,
is
c{v) £
weakly con-
another market feasible pair,
> U
{x [u)
,
e(j/)) for all
u G
J\f
J\f.
next provide an example with nonseparable state utility function in which the equilib-
\^'e
is
not constrained optimal, as implied by
optimal.
and 8
{p,[c{u)
the resource constraints in (13) hold.
market
,x{v),e
pair,
with strict inequality for a positive measure of f E
rium
=
A
Definition
The example shows
in part
Theorem
but which
2,
is
weakly constrained
that the inefficiency of equihbrium estabhshed in
Theorems
2
stems from the strong notion of optimality which gives the social planner a
technological advantage in monitoring.
Exciniple 3 (Nonseparable Preferences,
Example
1
Weak
Optimality). Consider the same setup as in
but assume, additionally, that workers are heterogeneous in their preferences. In
particular, there are
two types of workers, denoted by Tj and T2, of equal measure,
state utility functions of type Ti
u{xs,e ,Ti)
The
and T2 workers are respectively given by:
=
In
a;^
+
f
-
-I-
(1
-
=
In
J
((- +
29
{I
~ eU
xl
and
x^
e)
I
u{xs,e;T2)
1/2.
+
J
x^
(A.l)
.
Let
=
(p^
l^p^
=
=w
1, (c [Ti]
[Ti\
,
X
[Tj]
e [T'j])^grj
,
2}
denote an equilibrium price and
J
The
contract allocation pair in which workers of the same type accept the same contract.
equilibrium allocation for type Tj workers
(17)],
while the equilibrium allocation for type T2 workers
which the consumption of goods
1
and
Example
characterized exactly as in
is
Eq.
[cf.
the mirror image allocation in
is
We claim
2 are reversed.
1
that this equilibrium
is
weakly
constrained optimal.
Example
First consider the Pareto improving allocation in
monitor consumption), given by
social planner that can fully
by a
(the allocation chosen
1
e [Ti]
=
for
1
each
i
e
and
{1, 2},
the consumption allocations:
For type Tj
:
For type T2
:
xi
[Ti]
=
xj [Ti]
=
x\
[T2]
=
x] [T2]
=
^
and ±1
We claim that the corresponding contract allocation,
is
compatible
=
=
=
[T2]
=
[Ti]
{w
for the
type Tj workers only
<
if p'^
However,
j.
It
can make the contract allocation {c
remains to show that there
which
location
pair
reach
contradiction,
[p^
=
a
l,p'^,{'w[Ti]
G {1,2}.^^
=
is
a
is
,
there
is
>
p"
c [T2]}
ff
[Ti]
such
a
x[Ti]p,x[Ti] ,e[Ti])^^,^r,\)-
over
price
It
[T2]
=
is
=
[Ti]
(A.2)
0,
^
=
x [T]
A.3)
(
x [T] e [7^])}^^^
p,
,
normalized to
1).
Given
incentive
is
similarly, the allocation in {A.3)
Hence, there
2.
market
no other market
improvement
Pareto
that
[Ti]
if
xf
the allocation in (A.2)
1),
incentive compatible for the type T2 workers only
level that
i
{c
[Ti]
not market feasible for any relative price level p^ (recall that p^
the state utility function in (A.l) (and e[Ti]
is
and xl
and
no
relative price
feasible, verifying the claim.
price
feasible
the
is
and contract
al-
Suppose,
to
denoted
by
equilibrium.
allocation
pair,
can be seen that e
[T,]
=
each
for
1
In view of the conversion technology, the resource constraints can be reduced
to a single constraint:
,
EIE\E
i6{l,2}
(A.4)
-f[^]<^-"
,;
ge{l,2}
s&{l,h}
consume good
First consider the case p^ £ (51 2), so that type Ti workers always choose to
and type T2 workers always choose
to
consume good
2, i.e.,
xf
[Ti]
=
for
i
^
g.
1
Then, the
incentive compatibility constraints can be written as:
hn{ii[T,])
V
"^ ^J/
+
l-ln{xj[T,])
V
^'^
>
-
In^^^'f^^^
\2 p2
lln{xl[T2])
+ hn{xf[T2])
>
lnf^x?[T2]p2
2
More
specifically, for the case in
'
2
'
^
which only one type works,
a Pareto improvement without violating the resource constraints.
30
it
can be seen that
it
is
not possible to attain
2},
,
These constraints can be simplified to
s^UTil^Q
Note
(18)
also that this allocation
-(17))
if
only
i
with
that
2
a Pareto improvement over the equihbrium allocation
is
Eqs.
(cf.
-,
.
.
In {xl [T,])
+
^ In
some
[x] [T,])
>
E {1,2}.
i
bound on the amount
In
It
(^^tt^^ for each
i
e {1,2}
(A.6)
can be seen that conditions {A. 5) and {A.6)
of resources that need to be spent on each type workers,
is:
xUT,]+xJlT,]
>
(^±+^-^^ Inn
-2,rn
^
f ^P2
strict inequality for
,rr..
some
,
i
£
{1, 2}.
3
2p2
2^ +
2 \
Combining the
constraint (A. 4) (along with the fact that if
On
4M-.9
.
-2
with
„„,
if:
strict inequality for
establish a lower
1
[Ti]
—
l
i =/=
two inequahties with the resource
g),
we have:
13
2
3p2
+ -r +
6
last
iov
and
3^<y
.
the other hand, using the arithmetic-mean geometric-mean inequality,
3
3p2
2p2
2
?^ + J_
3
>
>
2,
inequalities with Eq.
can be obtained
for the case in
3p2
2P2
2
^2p2
2
3
3p2
=
we
have:
3 and
4
2.
3p2
Combining these
h
(^-^^
^
3'
(A. 7) yields a contradiction.
A
similar contradiction
which p2 ^ [5^2], proving that the equilibrium
is
weakly
constrained optimal.
The key
(A. 5).
intuition behind this result
Note that a high
2),
but
it
captured by the incentive compatibihty constraints,
level of the relative price p2 relaxes
straints for the type Ti workers
good
is
the incentive compatibility con-
(who have a greater temxptation to shirk when they consume
also simultaneously tightens the incentive compatibility constraints for the
type T2 workers (who have a greater temptation to shirk when they consume good
the Pareto improving allocation constructed in Example
price vector,
and the equilibrium
is
1
1).
cannot be decentralized with any
weakly constrained optimal despite the fact that
constrained optimal.
31
Hence,
it is
not
A. 2
Proofs
This appendix presents proofs of the results in the main text.
A. 2.1
Proofs for Section 2
We
show that Assumptions A2 and A3 imply the following lemma, which we use
first
in
the
subsequent analysis.
Lemma
(Local Transferability) Suppose Assumptions A1-A3 hold and consider an in-
2.
centive compatible contract, c
c+,c^ S
C\p)
=
{w,x,e) £ C\p). Then, for each e
>
contracts
0, there exists
such that
V{c^,p)>V{c)-€,
tt{c-,p)>tt{c,p),
(A.8)
and
V{c+,p)>V{c,p),
7T{c+,p)>TT{c,p)-e.
(A.9)
;
Proof of Lemma
We first
2.
the subsequent proofs. For each non-zero consumption vector xf^
6 K, we define w^ (x^^,p,
level ts
and some of
define a transfer function that will be useful in this
t^)
G
R
optimum
as the
,
price vector p,
and transfer
value and x^' (a;^\p,
as the
is)
solution of the following strictly convex optimization problem:
wl{x<^\p,ts)= min xf>^>
subject to
That
is,
w]
(xf^,p,ts) describes the
u'^'(xf')
is
interval (-^,5) such that
w]
over this interval,
vector
t
=
{ts)seS'
(xgi (xgi,^, ts))
We
and a
that
exists
^^
(5
>
is
ts,
is
(xf')
+ i^.
level necessary to increase the state s
given that the current consumption
that, since xj^
(^xf\p,ts)
^1^° define
is
well defined for each
>
2.
0.
(xf^',p, O)
tg
[x'^\p,t)
=
xf^ and
€ [-6,6). Note also that,
continuous and strictly increasing in
W''^
is
non-zero, there exists a sufficiently small
Given the transfer
tg.
(w] (xf^,p,ts))g and
the corresponding transfer functions over
Lemma
positive scalar e
we have w]
w]
Note
(^xf\p,ts)
o as
next prove
p.
u'^i
minimum wage
utiUty obtained from the goods in Gi by
the current price vector
=
(A.IO)
,
x"^' [x^'^,p,t)
=
all states.
Consider an incentive compatible contract c
=
C
{w,x,e) G
We first construct a contract c+ that satisfies Eq. {A.9).
= w] since c G C\p). Then, from the continuity of w^ (),
(p)
Note
there
such that
Wj
(if',p,
(5)
e
(u;],i(;s
-he) for each
(A. 11)
s.
_
32
By
w^
definition of
() (and incentive compatibility of
vP'
(icfi
Define the transfer vector
satisfies
By
=
{ts
vP' (xf 1)
+
(5,
for
S for each s). Since c
have
each
is
e S.
s
(A.12)
and u ()
incentive compatible
It
(8),
'
'
i
)
(^.11), this contract costs the firm at
Eq.
satisfies
=
{S)
=
also
Eq. (^-12) imphes that e remains incentive compatible after the utility
\
f (w^ (x^i p t{5)) u;^\(i})
follows that c+ =
^^ incentive compatible.
,_q^ \ q^^
, ,r/\'
g\Gi\
condition
transfer t{S).
t
{x^\p,5))
we
c)
most
Eq. (A. 9). Next recall that Assumptions
more than the contract
e
A2 and A3
ensure
u'l
>
Thus,
c.
each
for
a similar argument establishes the existence of a contract c_ that satisfies Eq.
c_|_
Hence,
s.
This
{A.8).
completes the proof of the lemma.
Proof of Proposition
(p,
[w
{i')
,x{h'),e
(i^)]i,g_/v)
ond claim holds holds by
7i"(c(i^),p)
definition of equilibrium.
>
all
is
0,
measure
that
set A/"*
in the constraint set of
of positive
A/^
=
{i^
is,
To prove the
the contract c{i')
measure zero
J
E Af*
Lemma
[i^)
= j}
C Af and
problem
measure and the
I
Applying
of
the
Let
proposition.
first
claim,
note that firms
first
is
in the constraint set of
We
set).
be accepted).
problem
claim that c{v) solves problem
but a measure zero set of workers. Suppose, to reach a contradiction, that there
exists a positive
both
part
if
profits (by offering contracts that will not
(10) for all u (except potentially a
(10) for
the only
denote an equilibrium price and contract allocation pair. The sec-
can always guarantee themselves
This implies
consider
First
1.
is
contracts {c{i'))^^j^, such that c{i') and c{u) are
(10),
set of firms
but V{c(i/),p)
2 to this contract for e
Since
C
J\f*
Af
there exists a firm j such that the set
is finite,
of positive measure.
> V{c{v),p).
Consider a contract
= V {c{i')
,p)
— V
{c,p)
>
0,
c(;y),
with
i^
£
TV-'.
there exists another
incentive compatible contract c_ {v) such that
V{c-iv),p)>V{ciu),p) andTTic_{u),p)>TT[ciiy),p)>0.
Then, we claim that another firm
contract offers to the workers in
workers v €
(cf.
Af-'
switch to firm
j'.
j'
Af-'
^
to (c_ (:^))j^g^j.
Note
Since each contract c_
{i^)
by changing
that, after this change,
makes the firm
all
its
the
positive profits
(A. 13)), the expected profits of firm j' strictly increase after this deviation, proving our
claim. This contradicts the fact that firm
proof for the only
We
for
j can strictly increase its profits
(A. 13)
if
j'
maximizes
profits in equilibrium, completing the
part of the proposition.
next prove the claim in the proposition that the constraint T:{{w,x,e) ,p)
any solution {w,x,e) to problem
(10).
>
binds
Suppose, to reach a contradiction, that the contract
33
,
c
=
{w,x,e)
is
a solution to problem (10) and
satisfies 7r(c,p)
exists another incentive compatible contract c+ such that
This contradicts the fact that
c is a solution to
problem
n
>
>
(c_|_^p)
(10),
Lemma
Then, by
0.
and
2,
there
V {c+,p) > V {c,p).
showing that the
profit constraint
binds.
Next consider the
price
if
part of the proposition.
and allocation system that
satisfies
Let p and [w {v) ,x (v) ,e{iy)]^^j^ be a
the two claims of the proposition.
ture a symmetric equilibrium in which every firm offers every worker the
=
{c{u,j)
i.e.,
[w{u) ,x{i')
Given these
,e{L')])^i^j^ -^j.
left
Suppose, to reach a contradic-
is
strictly positive profits
by offering a collection of
C
M with positive measure
incentive compatible contracts [c{y,j')]^^j^- Then, there exists Af*
,,,,,.
-
J
(u)
=
the deviation by firm
j' after
tt{c{i',j'),p) >n{c{iy),p)
=
Since the worker y e Af* prefers the contract offered by
,.:
•
c+
{i^,j')
and equations {A.14) and (A.15), there
2
,
and
(A.14)
f
over the contract
V{c{u,j'),p) >l/(c(i^),p).
.
j'
0.
.
By Lemma
opti-
offers are optimal.
can make
such that for each v € A^*
is
by assumption. Hence, we are
with by proving that firms' contract
tion, that there exists a firm j' that
same contract,
any worker strategy
offers,
conjec-
satisfied
mal. Moreover, the goods market clearing condition
only
We
exists
c(i^),
we have
,--
„
(A.15)
,
an incentive compatible contract
such that
tt{c+{u,j'),p)>0, &ndV{c+{u,j'),p)>V{c{iy),p).
It
follows that c(iv)
is
not a solution to problem (10) for v 6 Af*, which
is
a contradiction.
This shows that the price vector and contract allocations constructed above constitute an
equilibrium, completing the proof for the
We
next show that Assumptions
if
part of the proposition.
A3 and A4 imply
the following lemma, which
we need
to
establish the existence of equilibrium.
Lemma
A4
3.
(Continuity of Incentive Compatible Contracts) Suppose Assumptions Al-
hold Then,
the correspondence
incentive compatible contract c g
contracts c[n\ £
Since
C^
C\p\n]) such
(p) is also
C\p)
is
lower hemicontinuous in
C\p) and any
that c\n]
—
>
sequence p
[n]
—
>
p.
That
p, there exists a
is,
for any
sequence of
c.
upper hemicontinuous.
Lemma
p.
34
3 establishes the continuity of
C^
[p) in
,
Lemma
Proof of
Consider the price vector
3.
{w,x,e), and a sequence {p[n]}^i
n and a contract
c [n]
=
(x
[n]
e,
,
—
»
We
p.
w [n])
=
an incentive compatible contract c
p,
claim that, for any e
>
0,
there exists an index
such that
'
c{n\eC\p{n]) and w[n]€B{w, e).
Given
this claim, a
—
c[n]
»
To prove the claim
transfer vector
this vector
i,
t
s
(A.16)
sequence {c[n]}^Q can be constructed such that c[n] G C^{p[n]) and
C\p)
thus follows that
It
c.
'
is
lower hemicontinuous.
Assumption A4 implies that there
in (A. 16), first note that
such that Eq. (12) holds for the
M''^'
note that Eq.
and the vector
effort level e
(12) also holds for the transfer vector
where
-qt,
ry
exists a
Given
t.
>
an
is
arbitrary positive scalar.
Next consider the function w^ () defined
and
w^
be
since
w^
(a;^',p, Pj
(i'-'^,0,p)
e
B
=
strictly positive in
U (x {w,p, e)
the function
\\p[n]
—
<
p\\
5
<
\\T]t\\
=
s
is
(by Assumptions
(tf^e) nIR++ for each p and
sufficiently small so that
which
>
u.'^
view of Eq.
,
[n]
=
=
<
||i||
,
5
and
,e)
constructed
(x<^i (i<^i
,
p
[n]
,
<
{w^ {x^' ,p[n\,
that the effort level
^
e
such that
p||
<
Let
S.
77
(A.17)
y'?]f,gs(e),
p, there exists a sufficiently large
7]t)
,
,e)
+ e/2,
<
y(c,p)
>
U {x{w,p,e)
t/(x(u;,p[n]
and
rjt
x'^\^'
qt)
V {c,p)
and
n such that
and
,e)
~
e/2.
)
,
e)
,
e)
we
+
£ for all e e £; \ {e}
define
the
and we claim that
c[n]
p
[n]
,
(A.18)
.
c [n]
=
(^.16).
By
contract
\
satisfies
,
incentive compatible given the wages
exists e
—
f,
.
w^^^^}) /
construction of x[n] and w[n\ (and 5), we have that w\n\ £ B{w^,e).
[n]
||p
>
Since the indirect utility function
(12).
e
w
continuous in p and
is
these two inequahties jointly imply
e),
U {x{w,p[n],e)
X
max
()
and
V {c,p) = U {x {w,p, e)
the
,,
w^
A3), there exists J
that satisfies
i
continuous in
U{x{w,p[n],e)
Given
Since
A2 and
and define
y2''ltsqs{e)-
e) are
,
S
V{c,p[n])
Since
in (A. 10).
e is incentive
w [n]
and the
compatible.
effort level e.
Moreover, x\n\
is
Hence, we only need to show
Suppose, to reach a contradiction, that there
such that
t/(x(u)[n],jD[n],e),e)
>
35
;7(x(iu[n],p[n],e),e).
'
(A.19)
.
w [n]
Note that the construction of
Gi-
Note
differs
from condition
also that,
from
(8),
w
.
only for the wages for the monitoring subset
the consumption choice for goods within Gi does
not affect the consumption choice for the goods within other monitoring subsets. This implies
^G\Gi ^^
j^j
^p
j^j
^
g^
_
-j^GXGi
the definition of the function
(^yj^p
w^
()
Jtj]
^
e) for
any
These observations, along with
effort level e.
Eq. (A. 10)), further imply that:
(cf.
U {yi{w[n],p[n],e),e) =
+
'^r]tsqs{e)
t/
(x(u;,p[n] ,e)
e)
,
565
Considering this equality for e G
^rjtsqsie)
{e, e},
+ U {y.{w,p[n]
and plugging
,e) ,e)
> ^T]tsgs{e) +
t^
(x (u;,p[n] ,e)
,
e)
ses
seS
Combining
in the inequality (A. 19) implies:
this inequality
with the inequality in {A. 18), we have
Y^ T]tsqs
(e)
> Y^ rjtsqs
seS
(e)
-
e.
ses
This inequality yields a contradiction to the definition of e in (A. 17), proving that the effort
level e
is
incentive compatible. This shows that c
the proof of the lemma.
Proof of Theorem
to
.
The key
1.
p.
We establish
the local transferabihty condition
Let {{wn, ^n, £n)
that {w, x,e) €
,,,.
,
for the
problem
(cf.
Lemma
=
{w,x,e) and
satisfies the constraints of
2,
lim p„
problem (10)
Lemma
(cf.
[p),
3)
is
upper
is
along with
S (pn)
each n and
for
= p.
for the price vector p.
We
claim
to reach a contradiction, that there exists {w, x, e) that satisfies
V {{w,x,e) ,p) > V {{w,x,e) ,p).
(uJ, x, e)
which
strictly,
satisfies
the
,
7r((u;,x,e),p)
and which
C\p)
there exists another incentive compatible contract
non-zero profit constraint
S
denoted by
that {w„, x„, e„) £
the constraints of problem (10) and that yields the worker
By Lemma
.
,
2).
)Pn}^i denote a sequence such
S (p). Suppose,
(10),
this using the continuity of
lim {wn,Xn,en)
Note that {w,x,e)
the claim in (A. 16) and completes
step in establishing the existence of the equilibrium
show that the solution correspondence
hemicontinuous in
[n] satisfies
>
(A. 20)
0,
yields a strictly greater utihty for the worker,
V{{w,x,e),p)>V{{w,x,e),p).
36
'
);;.-
"
(A.21)
,
Since {w,x,e)
C^
(wJ„,x„,e„) 6
and
Lemma
C\p), by
£
(pn) for each n.
Since
there exists
^ (•)
and
7r(-)
>0
TT{{Wn,Xn,en) ,Pn)
since {Tu„,Xri,en) €
imphes that 5
:
(A. 20)
C^
>V {{Wn,Xn,en) ,Pn)-
{pn), these inequalities contradict the fact that (Tj;„,x„,en) is
S
follows that [w,x,e) 6
It
a
which
{p),
upper hemicontinuous.
{p) is
rest of the
D
dence
{w,x,e) such that
»
are continuous functions, Eqs.
and V{{Wn,Xn, en), Pn)
solution to problem (10) given the price vector pn-
The
—
(tZ;n,5„,e„)
imply that there exists a sufficiently large n such that
(-4.21)
But
3,
proof follows standard arguments. Consider the excess
{1} X r!^
RI*^I
:z5
'
good
(recall that the price of
correspon-
normalized to
is
1
demand
given
1),
by:
=
n(r,]
^^'
/AAEse5(^.(^)-2/.)9.(e(^))di^6Kl^l
{
[
Since there
S
Since
We
ys
(p) is
<
Qo for each
(p)
s)
while the
<
(p)
good g tends
>
D^
since the supply of
0,
demand
since the supply of
to zero as p^
—
»
for
D [p)
^
good g
\
good g tends to
good g
oo. Therefore,
is
{w
{u)
,x[v) ,e
A. 2. 2
{v))^j^j^
clear.
\s
bounded from above
infinity as p^
—
Similarly,
0.
>
(since
zero, while the
demand
Kakutani's Fixed Point Theorem applies to this
(i/)
,x
(i/)
G
D (p).
By
S {p)
,e{u)) e
definition of
for
each u &
J\f
the price vector p and the contract allocations
1,
correspond to an equilibrium, completing the proof of the theorem.
2.
good g between
N*
Consider a worker v e
Since {x,e) does not feature
for
Proposition
upper hemicontinuous.
Proofs for Section 3
Proof of Theorem
MRS
By
'
convex valued.
is
also
bounded away from
there exists {w (v) ,x{i'),e {u))^i-j^such that {w
and the goods markets
^
is
economy, which implies that there exists a price vector p such that
D (p),
D [p)
demand correspondence
upper hemicontinuous, the correspondence
have, limps_^o
limp9_oo -D^
for
a continuum of workers, the
is
1
iwiv),x{u),eiiy))eS{p) ioie&chiyeM j'
\
full
states si
and denote her allocation by
insurance, there exists si,S2 G
and
S2 is not equal to
1.
We
S and
g G
will reallocate
G
c
=
[w, x, e).
such that the
the consumption
of good g across states so that the worker receives better insurance while her equilibrium effort
choice e remains eff'ort-incentive compatible.
Formally,
we claim that there
exists x^
\s\
G W_^
such that
9.(c)i|
= ^9=
(A.23)
(«)!?,
s€S
ses
e
and t/(^^\{^^i^e)
G
>
aigmaxU
eeE
t/(x,e).
37
(x^''^^\x^,e],
\
(A. 24)
J
(A.25)
Once we prove
this claim,
improving deviation
it
follows that there
an effort-incentive compatible and Pareto
is
each worker u G Af*, which implies that the equilibrium allocation
for
is
not constrained optimal.
To prove the
-^—
dill
that
(j4.25). Recall
we
claim,
Xo
''
show that there
first
c
"'
(
Suppose, without
1.
=/=
du{xsi,
du
e)
>
9
OXq^
For any e
>
consider the deviation vector v
0,
exists a deviation xf that satisfies (A. 23)
and
/9x
1
loss of generality, that
(x52, e
;j'
(A-26)
•
OXs2
G
[e]
R''^'
defined by Vs
[e]
=
for
any
^ {s\, S2}
s
and
r
^^
1
1
r
^^
1
Qsi (e)
where the constant
K=
—
^
—
°^
°^
'
^
by construction, the vector x^
=
x^
>
v
-\-
,
«™x
»
qs2 (e)
ensures that
<
e for
any
e
>
0.
Note that,
the resource constraints in (A. 23). Note
satisfies
[e]
\\v [e]\\
also that
U
(x^\^«> x»
,
+
t;
[e]
,
= J^ g^
e)
(e)
w (xf ^^^l xf
,
„
line considers
a
first
[e]
U
Eqs.
(x'^^'S',
(A. 28)
x^
+
and the inequahty
v[e] ,e)
> U
(x, e) for
e)
order Taylor expansion for the functions u {xg^ e)
,
and u{xs2,e), and the notation o{e) captures the residual which
From
,
aK(x,
,
'?^i ^^'l
.
where the second
,
^,
E56S9^(e)"(2^^'e)+
\
e)
au(x3,,e) e^y
^^
VS2 le;— gj|-—
qs^{e)
dxi^
^^^
/
-
+
in (A. 26),
each e e
it
satisfies
>
follows that there exists £
(0, e).
^^ =
lime_o
0.
such that
This proves our claim that there exists a
deviation x^ that satisfies (A. 23) and (A. 25).
We
next consider a sequence of vectors {v
such that £
f5
=:
x^
[n]
+ v [e
G (0,£) for each n and e[n]
[n]]
[n]
e
[n]]}^-i,
—
>
0.
also satisfies the effort-incentive
to reach a contradiction, that there
e
[e
E\{e} such
is
no such
n.
where {e [n]}^
We
is
a sequence of scalars
claim that there exists n such that
compatibihty constraint in (A. 24). Suppose,
Then,
for
each vector
i'[e[n]],
there exists
that
U
Since the space £'\ {e}
subsequence. Let e E
is
E\
(x^\^3\xs
finite
+ v[e[n]],e[n]) >U{x,e).
(A.29)
.
(and thus compact), the sequence {e [n]}^-[ has a convergent
{e} be a limit point of this sequence. Since hm„_^oo v
38
[e [n]]
=
0,
Eq.
.V
(A.29) implies
:
Note
U{x,e) >U{x,e).
.
that, since preferences are nonseparable at (x,e), there exists
(A.30)
.
s,m and
gi,g2 6
Gm
such
that
du{xs,e) /dxf
du{xs,e) /dxV-
_
~
du{xs,e)/dxf ^ du{xs,e)/dxf
where the
last
C
equality follows since c £
xl^'-^'^ (u;,,p,e) 7^ xi^''^^^
The
{p).
last
pSi
^'
equation further imphes that
This further implies
U {x{w,p,e) ,e) > U {x,e) > U {x,e)
where the
last inequality uses
contract {w, x, e)
x^
=
x^
+v[s
[n]]
is
This yields a contradiction to of the fact that the
(A.30).
incentive compatible.
satisfies
,
It
equations (A. 23)
follows that there exists a vector v
— (A. 25), which
[e [n]]
such that
completes the proof of the theorem.
Proofs for Section 4
A. 2. 3
The proof
of
Theorem
3 requires a preliminary step.
intuitively clear that,
if
preferences
then the indifference ciurves between any two goods in
satisfy the separability condition (19),
the same monitoring partition
It is
m should be independent of the effort choice
e.
The next lemma
formalizes this observation.
Lemma
subset
in
Gm
Suppose the separability condition (19) in Definition 4 holds for the monitoring
4.
Gm- Consider
such that the
a vector {xg, e) £
IRi|_
x E, and
let
xf"^ denote a reallocation of the goods
level of the utility is kept constant, i.e., suppose:
u(if-,xf\^"',e) =u(a;„e).
Then, this reallocation keeps the
cf ""
Lemma
Proof of
4.
We
first
,
level of utility
xf \'^'"
,
e)
constant also for any other effort
=u{xs,
e)
for any e e E.
level, i.e.:
(A. 31)
claim that, under the separability condition (19), the partial
derivative of the utiUty admits the following representation:
^}^^i^ =
where
F
:
e'^' -^
R''^'"!
is
a vector valued function and h
positive scalar valued function.
To prove
(A.32)
h{is,e)F{xs),
this claim, fix
39
:
Rl^^l
x
some good
E ^ R++ is a strictly
g € Gm and define the
.
..
scalar valued function
du
(xo, e)
h{xs,e)
which
is
strictly positive.
Applying condition (19)
6 Gm,g), we have that the
for all pairs [g
ratio,
is
independent of
du{xs,e) /dxf'^
du{xs,e) /dif""
du{xs,e) /dxi
h{xs,e)
Hence,
e.
can be denoted by a vector valued function F{xs)-
it
This
completes the proof of the claim in (A. 32).
Next, to prove the lemma, note that there exists a continuously differentiable function
xf"*
:
[0, 1]
-^ 1+"" which satisfies
u
Note that xf "*
(i)
("xf "
(i)
,
xf"
xf
(0)
^^'^
=
=u {x„ e)
,e]
represents a curve that
lies
=
xf"", xf"' (1)
for
if'",
each
t
e
and
(A.33)
[0, 1]
inside the indifference surface (which
is
the higher
dimensional analogue of an indifference curve). Totally differentiating Eq. (j4.33) with respect
to
t,
we have
^
9^i(xf-(i),xf\^",e)'dxG'"(t)
L
^f^_W =
L
.
for
each
t
e
[0, 1]
dt
dx?"^
'
'
,
Plugging in the representation in (A. 32), the previous equality implies:
h (xfSince
F
/i(-)
(xf"
(t)
(i)
xf \°"* y
h (xf-
which
is
the
an arbitrary
xf \^-,
F
e)
(xf-
(i)
,
xf \^'-y ^^E£_lW ^
Q f^^ g^^j^
g
J
[Q^ ^]
^^34^
_
a strictly positive scalar valued function, the previous equality implies that
is
,
,
(t)
same
e
,
^^V^ = 0xf \^-
as Eq.
,
e)
F
This further imphes that
(x?-
it)
,
xf \^-)'
,
^^£^ =
.
each tG[0,l],
for
(A. 34), except for the fact that the effort level,
e,
is
replaced by
E E. Using the representation in (A. 32) one more time, the previous equality
imphes
du (xf -
[t)
,
x?^*^""
V
,
^^Gm u)
i ^^s {^)
e)
^
f^^ g^^j^
dt
dx?"'
t
e
[0, 1]
^
'
'
Integrating this equation and using the fundamental theorem of calculus,
(A. 31), completing the proof of the
We
next use this
lemma
lemma.
to provide a proof of
40
Theorem
3.
we
establish Eq.
Proof of Theorem
By
3.
the equilibrium conditions, [x {v)
Assume, to reach a contradiction, that
exists
an incentive feasible allocation
[x{i/) ,e (i^)]^^/^ is
\x (v)
e
,
{i^)]i,^j>j-
,
e
{i^)]j^qj^/ is
incentive feasible.
constrained suboptimal. Then, there
such that
'
U{xiu),eiu))>U{xiu),e{v))
with
strict inequality for
Consider v e
a positive measure of
N such that the inequality
We
notation for convenience.
will
of the indirect problem (10), that
i/
G
(A.35)
A/".
(A.35) holds strictly, and drop the
show that the allocation
i/'s
from the
the profit constraint
(i, e) violates
we claim
is,
Y,qs{e){ys-Xs)p<Q.
(A.36)
^
ses
And
similarly,
we claim that the same
holds weakly. Once
of the
we
establish the claim in (A.36), the result follows
welfare theorem.
first
In particular, integrating Eq.
contradiction to the fact that \x{v)
that the equilibrium
is
To prove the claim
inequality holds weakly whenever the inequahty in (A.35)
,
e
(i^)]i,g_A.^
from the standard proof
(A.36) over
&\\
M yields
u £
satisfies the resource constraints,
a
which proves
constrained optimal.
we
in (A.36),
construct a contract c
will
=
{w,x,e) which
satisfies
the
following three properties:
(PI) c
is
incentive compatible.
(P2) c yields the worker the same utility than the allocation
(P3) c costs the firm less than the allocation
(x, e).
(x, e), i.e.,
S2qs{e)xsP<^qs{e-)^sPOnce we construct a contract
(PI), the contract c satisfies
(11).
By
all
c
(A. 37)
seS
ses
with these properties, we have the following impUcations.
By
the constraints of problem (10) except for the profit constraint
(P2), the contract c yields the worker greater utihty than the equilibrium contract.
Since the equilibrium contract solves problem (10), the contract c must violate the remaining
constraint of problem (10), that
is,
violates the profit constraint.
it
By
(P3), the contract c
costs the firm less than the allocation (x, e), which in turn implies that (x, e) also violates the
profit constraint.
we
Hence,
properties
limit
sively.
of
This shows the claim
are
left
(P1)-(P3).
a
sequence
In particular,
with
We
in
constructing
construct
will
{w[n] ,x[n]}^_Q,
let
Eq. (A.36), completing the proof of the theorem.
(ti) [0]
,
x
[0]
)
a
contract
the
{w
=
41
=
sequence
that
(ii),x,e)
wage and allocation
where the
=
c
will
xp,x) denote the
be
pair,
satisfies
{w,x)
as
constructed
initial
vector.
the
the
recur-
Suppose
.
{{w
[0]
,
X
[0])
tion of {w
[n]
,
{w
..,
,
X
[n
[n]).
x[n]^= Cxf'",i[nvalue and Scf""
is
-
Let
1]
,
X
[n
m€
-
{1,
l]f\'^'"')
I])} is
and w[n]^= ('w^,u)
-
[n
llf^^""^), where
n. For
w™
is
each
the
define
s,
minimum
u
x^^p*^"
(x'^'^
,x[n-
each step, the vector, {w
at
and consider the construc-
the unique solution to the following strictly convex optimization problem:
subject to
is,
1,
\M\} denote the modulo \M\ value of
..,
min
That
>
constructed for n
(A.38)
1]^\'^"'
,x
[n]^
,
= u{x[n -
e")
(A.39)
l]^,e)
constructed by reallocating the goods
[n]^), is
within one partition, Gm, in a way to minimize the costs while providing the worker with
the
same
The
utility as before.
partitions are subject to this operation one at a time
an order. Once we operate over
in
all
we
partitions,
start the process over (which
We claim that
captured above by taking the modulo \M\ value of n).
converges to a vector
We
first
{'w.,x),
show that the sequence
the wage for each partition,
c=
and that the contract
{iD [n]
,x
(u) [n]^
,
[n]^
i
""
{w [?^]™}^o
for
each
constant
it is
m and
s,
w [n]^
conditions:
w [n]^,
bounded below by
it is
"*
[n]^
•
-
>
A
du x
H7 ^ dxi
[n] ^
{
}^Jg'
with equality
[n],
partition,
weakly decreases
0.
This means that,
to
problem
(j4.38) satisfies the first
order
'
P^
w
satisfies (P1)-(P3).
has a unique limit point which we denote by w^. This further implies
that w\n] -^ w. Next, note that, the solution x
where A [n]^
,x[n]}
an updating does not occur). In particular,
if
a decreasing sequence. Moreover,
is
[n]
updated once every \M\ turns. Moreover,
is
problem {A.38) shows that the wage corresponding to each
each time an updating occurs (and
{w
formally
Note that the allocations and
converges.
[n]}
j,
{w,x,e)
the sequence
is
and
is
if
,
'
eo)
"^
if""
for
>
'
each
s,
m, and g £ Gm,
0,
a strictly positive Lagrange multiplier for each
the solutions x
[n]
(A.40)
s
and m. Hence, given the wages
are uniquely characterized by the conditions in (A. 40) along with the
equations
w [n]^ =
X [n]f " p^"'
for
each
which hold since the minimum of the problem {A.38)
{A.40)
—
(A.41) are continuous in
to a vector x.
The
ib [n],
limiting vector, x,
is
and
since
w
[n]
s
is
and m,
:
'
attained.
Since the equations in
-^ w, the solutions x
the solution to Eqs. (A.40)
—
(A.41)
[n]
also converge
(A.41) corresponding to
the limiting wages w. This establishes that the sequence we have constructed converges to a
vector, {wjx).
42
We next show that the contract c =
We
{w, x, e) satisfies (PI), that
break this argument into two steps.
incentive compatible, that
'"
Vjg^
(e)
u
(fs, e)
> 2~]9s
,
(e)
e)
"(^s,
for
effort-
is
each e € £.
(A. 42)
utility is
compatibiUty condition
c satisfies the stronger incentive
of the claim in (A. 42) rehes on
imphes that the
that the allocation (f e)
ses
We will then establish that the contract
The proof
compatible.
is,
ses
(4).
we claim
First,
c is incentive
is,
Lemma
4.
In particular, note that Eq. (A. 39)
preserved at each step of the above construction, that
u{x
[n]^
,
e)
= u{x[n —
1]^
,
e)
each
for
s.
is:
,
Moreover, note that, at each step, the above operation reallocates the goods within exactly
one monitoring subset Gm- That
is,
the allocations x
(potentially) for the allocations x [n]^ ". Hence,
u{x\n]^
,e)
= u{x[n —
This further implies that the limiting
that
Lemma
1]^ ,e)
utility is
[n]^
for
and x[n —
4 applies
each
equal to the
5
and
1]^
same except
are the
and shows that
e
G
£'.
initial utility for
any
effort level e,
is:
=
u{xs,e)
Since the allocation
(x, e) is effort-incentive
(x, e) is also effort-incentive
We
u{xs,e) for each s and e e
compatible, Eq. (A. 43) implies that the allocation
compatible, proving the claim in (A. 42).
next prove that the contract c
is
incentive compatible. Given wages
choice e £ E, the worker's consumption choice satisfies the following
du{xs,e)
^3
dx;
< 7™p^
with equahty
where
7^
is
if
for
each s,m,
xf "> >
is
gG
first
w
and some
effort
order conditions:
(A. 44)
G"",
0,
a positive Lagrange multiplier for each
consumption choice
(A. 43)
£^.
s
and m.
In particular, the workers'
uniquely determined by the conditions in {A. 44) along with the budget
constraints:
lii^
Note that the
(A. 40),
e,
and
is
—
for
each
s
and m.
(A. 45)
order conditions in {A. 44) are identical to the
and Eq. (A. 41)
to Eqs. (A. 40)
choice
first
= xf ""p*^"
is
identical to Eq.
(A. 45).
(A. 41). Hence, the worker's
equal to
x.
first
Recall also that x
consumption choice
is
is
order conditions in
the unique solution
independent of her
effort
This imphes that the effort-incentive compatibility condition (A. 42)
imphes the incentive compatibihty of the contract
43
c,
estabUshing (PI).
We
next show that c
utility to the
is
(P2) and (P3). Note that, by construction, c yields the
satisfies
worker as the allocation (x,e), establishing (P2). Recall also that
w=
a decreasing sequence, which implies that
firm less than the allocation (x, e) and Eq.
the proof of
A. 2. 4
Theorem
w [0] =
["-J^j^o
xp. Hence, contract c costs the
This completes
(A. 37) holds, establishing (P3).
3.
Proofs for Section 5
Proof of Theorem
if
xp <
{it'
same
and only
if
We
4.
(p, [fJ-uJu^Af)
Given
the space,
2,
V
iC
argument
this result, the
(p)), also satisfies the
in the
°^ ^^ equilibrium
^^ P^'-*
^
conditions 1-2 of the theorem are satisfied. Since the space,
transferability condition in
condition.
claim that the pair,
first
C^
(p), satisfies
the
analogous transferability
proof of Proposition 10 applies unchanged
with stochastic contracts, proving the claim.
We
next show that an equilibrium exists. First,
problem
(25)
compact. Since the objective function
is
solution to problem (25) always exists. Next,
S^
denoted by
(p)
C V
with weak* topology).
V
(C^
to
Theorem
(p))
:
M'"^!
1
[C^
^ V (C)
is
rest of the
D^p)
Note
that, for
C^
3,
(p) is
=
D
:
(and thus continuous) in
linear
we claim that the
S
(p) is
(recall that
V (C)
S^
upper hemicontinuous. Recall also that
e
R|^
=5
'
IRl'^l
For a given price
M
any collection of allocations,
G
(p) is
convex
p, define the excess
\ji^,
e
S^
(p)]
,
the excess
^
'
]
demand
in (A. 47) is
.
V {C)
]1
that
by
I
is
S^
.-
^^g
"
the average measure defined by
fifc'j
Note that
endowed
hnear.
JJ\f J(w,x,e)
jl
is
p. It follows
equivalent to
where
a
Then, a similar argument
^MkH^')M-),e{-))^sss(''s{y)'ys)qs{e{u))dlL^dv
p^ £ 5^ (p) for each v ^
[
\
/i,
solution set to problem (25),
a continuous correspondence in
proof follows standard argimients.
demand correspondence
is
also a continuous correspondence in p.
establishes that
is
can be seen that the constraint set of
upper hemicontinuous in p
(p)), is
By Lemma
valued since problem (10)
The
it
(p), since
Jl^,
G
=
f
S^
(p) for
p.^
(c) dv
for
each
each v and
S^
C eB{C).
(p) is
convex valued. Hence, problem
(A. 46) can equivalently be written as:
Dip)
=
\f
,
5](x3-y.)g.(e)d/i|/ie5«(p)l.
44
(A.47)
.
S^
Since
(p) is
upper hemicontinuous
Theorem
saxne arguments in the proof of
proof of Theorem
As
5.
compatible allocation,
t),
A* Hsupp
and that
closed)
show that an equiUbrium
1
77
Theorem
in the proof of
exists,
in p.
Then, the
completing the
=
t]
show that there
will
is
an incentive
My|(x,e)-
(77),
and note that A**
=
{A*)
(^**)
we
2,
that satisfies the resource constraints and improves the utility of the
worker over the equilibrium allocation,
=
upper hemicontinuous
also
is
4.
Proof of Theorem
Let A**
D (p)
in p,
rj
>
Let
0.
compact
is
Theorem
compact and supp
(77) is
Then, consider the deviation
e).
>
In particular, for each £
2.
is
e A** and note that Definition 7 imphes that
(x, e)
preferences are nonseparable at the deterministic allocation {x,
coiistructed in the proof of
A*
(since
v
0, let
6
(x, e)]
[e
M'"^!
\
denote the vector constructed in (A. 27). Note that
\\v\e
(x,e)]||
<
£,
and that v[£
\
is
continuous in
-'/
each e
for
<
e
Theorem
Recall also that, by the proof of
(x, e).
'
+
fx^\^»^ x^
f/
(x, e)]
[e
I)
>
e)
,
I
and each (x,e) e A**. For each
e
(x,e)]
\
there exists s such that
2,
(A.48)
;7 (x, e)
e (0,5), define the function ({,£):
A
—*
A
with:
(x,e)
C\[g]
{x^\^3}^x9
C((x,e),£)-^
Note that C('i£)
+
I
if
v[s\
A\A**,
G
(x, e)
(x, e)]
,
e A**.
(x, e)
if
e)
a measurable function from {A,B{A)) to {A,B{A)), because the pertur-
is
bation function v[£
(x, e)] is
\
continuous in (x,e). Hence, given the probability measure
the function C('i-) induces another probability measure over {A,
6
77,
{A)) (the push-forward
measure), defined by:
','-1
=Til^C [A,
T][e]I^Aj
Note that
of
77
[e],
We
77 [e]
is
a stochastic allocation. Moreover, by equation (A.48) and by the definition
we have that U^
{rj [e])
sequence of scalars such that
supp
n,
By
77
(77 [e
[e [n]] is
(77)
and that
[n]]) is
.
U
is,
e
(0, e) for
each n and e
{77 [e
[n]
—
>
[7i]]}^j,
0.
where {e
We claim that
[n]}^_-^ is
a
there exists
n
not a subset of the set of deterministic incentive compatible allocations,
the construction of
A^ That
e [n]
the resource constraints (27).
satisfies
77 [s]
also incentive compatible. Suppose, to reach a contradiction, that for each
77
[e [n]],
that the perturbed vector
of
> U^
next consider the sequence of stochastic allocations
such that
A eB{A).
£)]. ioi each
(
x
there exists e
(x [nf^^^''
,
X [nY
^^^'
[n]
[ti]
a vector (x
this implies that there exists
,
e
£\
+ v[s [n]
x
[n]^
+
v[e
(x
[n]
[n\
,
e
[tt])]
,
[n]
e [n]
)
,
e
is
[tz])
A
.
G A** such
not an element
\
{e [n]} such that
(x
|
[71]
,
e [n])]
45
,
e [n])
>U{x\n],e
[n])
(A.49)
.
Since the space A** x
E
compact, the sequence of vectors,
is
convergent subsequence. Let
e 7^ e (since e
[n] 7^
e [n] for each n).
E he
6 A** x
{x, e, e)
Note
[n]}^j, has a
{(2; [n] ,e [n]) ,e
& Hmit point of this sequence and note that
also that Eq. (A. 49)
impUes
U{x,e) > U{x,e).
Since preferences are nonseparable at the deterministic allocation
Theorem
G A**, by the proof of
{x, e)
we have
2,
U {x{w =
xp,p, e) ,e)
> U
c=
This yields a contradiction to the fact that the contract,
ible (the contract c
=
{w
xp, x, e)
is
incentive
compat-
in the
support
incentive compatible since the allocation, {x, e) G A**,
is
of the equilibrium stochastic allocation,
Proof of Theorem
> U (x, e)
{x, e)
77
—
/^^|(x,e))-
This completes the proof of Theorem
Suppose to obtain a contradiction that
6.
is
('7^)|^gA/' is
5.
not constrained
optimal.
Then, there exists an effort-incentive compatible and feasible stochastic allocation
ilu)
such that
\u€J\f
with
strict inequality for
a positive measure of v G
Consider an allocation
drop the
e
rji^
from the notation
i^'s
M
for convenience.
profit constraint of the indirect
problem
/
J(w,x,e)eC
And similarly, we claim that
We
(25), that
(A.50)
J\f.
which the inequality
for
,-•:.,:,:•
-
i7^(r?J>t/«(77j
will
is,
in (A.50)
show that the
is
satisfied strictly,
allocation
77
and
violates the
we claim
X]9^(^^(2/s-a;s)p cR7<0.
V^g5
/
(A.51)
.',,
the same inequality holds weakly whenever the inequality in (A.51)
holds weakly. Once we estabhsh the claim in (A.51), the result follows from the standard proof
of the
welfare theorem.
first
In particular, integrating Eq.
(A.51) over
all
v G Af yields a
contradiction to the fact that {t}^)^^/^ satisfies the resource constraint (27), which proves that
the equilibrium
Consider
the proof of
where
is
(x, e)
constrained optimal.
£ supp
Theorem
3,
recall that (PI)
-
,
{fj)
and note that
(x, e)
there exists a contract c
<=^
c
(P3)
G C'
(p),
4=>
(P2)
{w
the construction in
[x, e]
,
X
[x, e]
,
e) is
Theorem
A-' since
{w,!,
-^^ U
e)
fj is
3,
incentive compatible.
(x,e)
=U
and
(x,e),
:;
can also be seen that the contract
a continuous function of (x,
46
e), for
each
(A.52)
•
,
seS
it
By
that satisfies properties (P1)-(P3),
^g,(e)x,p<^g,(e)x,p.
seS
Prom
=
C
(x, e)
£ supp
(?/).
c [x,
Extend
c
e]
[•]
=
to a
measurable function over
Note that
c
[•]
measure
bility
by defining
c [x, e]
over
B {A)),
(^4,
the mapping c
=
[•]
=
{ili
B {A))
a measurable mapping from {A,
is
t]
A
of
all
xp, x, e) for each (x, e) ^ supp
to (C,
B (C)).
{fj).
Hence, given the proba-
B (C))
induces a probability measure over (C,
(push- forward measure), defined by:
jl
Note that the support of
each
(x. e)
=
(C^
fi
in
is
(c-^
fj
C^
Hence, the stochastic contract
6 supp[f}).
is
problem (A. 50),
it
follows that
fi
(cV
is
jj.
e]
e
incentive compatible.
C'
Moreover,
yields the
fi
{p) for
worker
Since the equilibrium allocation
p..
violates the remaining constraint of (yl.50), that
(P3) and the definition of
more than the
proof of Theorem
e B
/z
is:
\^(}s{e){ys-Xs)p\dlL\(^^^)<Q.
/
costs the firm
C
satisfied strictly implies that
than the equilibrium allocation
strictly greater utility
By property
each
property (PI) imphes that c[x,
[p), since
property (P2) and the fact that (A. 50)
solves
for
(^C'j'^
allocation
6.
/i,
r\.
we
also have that the stochastic allocation y\ix^e)
This imphes the inequality (A. 51), completing the
.
.
Proofs for Section 6
A. 2. 5
Proof of
Lemma
We
1.
claim that U3^^^^^ {u)
first
>
U^^ {p,u) holds for
equiUbrixmi price vector p e P{u). Note that the solution to problem (28)
is
greater than the solution to problem (25) (because the equilibrium allocation
all
U
u e
and
always weakly
pA/
is
always in the constraint set of problem (28)). Since problems (28) and (29) are equivalent,
it
follows that f/^o„„er (u)
We
>
t/g^ {p, u),
next prove the lemma.
librium allocation,
Pi,\(x e)
tion, that ?7p^^„„,,(u)
i
>
proving the claim.
First consider the only
^^
[Mi/|(x,e)]
if
e-constrained optimal.
[uS,ip,u),U,^^{p,u)
+
Since
e].
part, that
Suppose,
is,
suppose the equi-
to reach a contradic-
U^nnerM >
U^q{p,u), we have
tion
[fii,]^^j^,
+ e. Let denote the solution to problem (29) and consider the allocadefined hy fj^ =
for each v. The allocation [f]^]^,^j^ is effort-incentive compatible
and
feasible,
and
^planner
i'^) -*
^eq
(Pi
u)
f]
fj
it
improves every worker's
that the equilibrium allocation
of the
[/J.i,|(x,e)]
^at
utility
i^
by more than
This contradicts the fact
constrained optimal, proving the only
if
part
lemma.
Next consider the
if
part, that
is,
suppose f^Q„„gr
pose, to reach a contradiction, that the equilibrium
exists
e.
an allocation,
proves the utility of
[fii^Ji^^j^,
all
which
is
is
('^)
^ [^Sj
47
for
^)
i
^4 (^' "") + ^]
•
^^P"
not constrained optimal. Then, there
effort-incentive compatible
workers by e (and strictly so
(Pi
and
feasible,
and which im-
a positive measure workers). Consider
the average allocation
is
f]
defined in Eq. (30), which satisfies the resource constraints, and which
incentive compatible since
Moreover, since
(29).
utility of all
^e^
(P' '^)
Proof of Theorem
W
is
{A
is
a Pareto improvement, the average allocation
+ ^'
is
a convex set. Hence,
We
7.
that
e,
first
{u)
=
<.{x,e)
A
e
(u)
is
Assumption A3
Lemma
3
is
is
for the
shows that A^
(p,
(e)
u
> S^ 9«
{x, e)
{u)
is
same function
(^)
^
u'-'^ (•),
Consider next the correspondence
U^^{u)
the correspondence
t/g
u.
is
the same problem.
A
similar
argument as
in the
(p,
u)
is
:
W
M
=?
It
(25),
and
V {^A^ [u)) is
U to V{A)).
a version of the
proof of Theorem
4,'
= ll
Then,
('") ^^
continuous in u.
is
(A.53)
-
We
in u.
To
see this, let
S^
claim that
{p,u)
C V {A
U^{p,u) denotes the optimal value
continuous correspondence of {p,u).
demand correspondence
By
)
of
U^ (p, u)
the
is
given by
upper hemicontinuous
in (p, u).
same
Y,{xs-ys)qs{e)dfi\~^eS^{p,u)y
upper hemicontinuous,
D {p, u)
is
also
over, note that the equilibrium price correspondence satisfies:
p (u) =
also a
defined by
recall that
the excess
U
Maximum
argument to the previous paragraph establishes that
a
.
follows that
(i.e.,
continuous
= {U^^{p,u)\pEP{u)],
is
Dip,u)
S^
U
and S{p,u)
a continuous function
Since
is
>
shows that the solution to problem (29)
upper hemicontinuous
denote the solution to problem
is
1
E
Next, since each u E
the equilibrium price correspondence defined in (32).
[u)
(where recall
an argument similar to the proof of
applies to problem (29)). This shows that ^^anner
is
e
^°^ ^^'^^ e
(^' ^)
This further implies that
upper hemicontinuous, and the optimal value
P [u)
U
a lower hemicontinuous correspondence of u.
a continuous correspondence of u.
recall that
follows that
(u):
an argument similar to the proof of Proposition
where
improves the
It
z.
problem
consider the set of deterministic
this, first
continuous correspondence of u (when viewed as a correspondence from
Theorem
+
u)
i)
continuous in u E
an upper hemicontinuous correspondence of
satisfies
{u)
A
yj qs
\
Note that A^
in the constraint set of
U^ (f)) > U^
claim that U3^^^^^ (u)
a metric space with the sup norm). To show
A^
is,
is
fj
which yields a contradiction, completing the proof of the lemma.
incentive compatible allocations,
A^
)
workers by strictly more than
>
^pianneT (")
\f]^]^
V
r!^i
|p e
48
j
e
D (p, u)
More-
.
D {p, u)
Since
is
is
upper hemicontinuous
in (p, u),
an upper hemicontinuous correspondence of
U^ {p, u)
is
it
has a closed graph, which imphes that
P (u)
Since
u.
continuous, Eq. (A. 53) imphes that Ug^ {u)
is
is
P (u)
upper hemicontinuous and
upper hemicontinuous, proving the
claim.
We
have thus estabUshed that US^^^^j.
(u)
is
a continuous function and U^^ (u)
uous correspondence of u e U. Next consider the values of these expressions
that any equihbrium of the economy S (u)
Lemma
1, it
follows that Ug^ {u)
is
is
constrained optimal by
a singleton and
it is
u
for
Theorem
=
6.
equal to US^^^^^^ (u), that
a contin-
is
u.
Then, by
is,
(A.54)
Ki^)={Uplanner{u)}.
Fix some e
||u
-
<
nil
>
0.
From the
Jpianner,
continuity of US^^^^^
(•),
Recall
>
there exists 6pianner
such that,
if
then
(A.55)
'
\UXnnerN-UXnneri^)\<£/^D
Similarly,
if ||U
-
u||
from the upper hemicontinuity of
<
SplanneT,
=
min
and Eq. {A.54), there
exists 6eq such that
then
Ue,
Let 5
t/g„ (•)
[Spianner, 5eq)
- U±^^,A^) <
5/2 for any t\, e
1/
J
(A.56)
(u)
and note that Eqs. (A.55) and (A.56) imply U^^^^^^^
{u)
-
Ueq
r>
for
any u e
B {u, S)
any economy 8^
Theorem
(u),
and any Ueq
J eg €
U gg
G U^q
with u G
B (u,6),
{u).
is
By Lemma
1, it
follows that
any equihbrium of
e-constrained optimal. This completes the proof of
7.
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