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HOLDUPS AND EFFICIENCY
WITH SEARCH FRICTIONS
Daron Acetnoglu
Robert Shimer
September 1998
massachusetts
Institute of
teclinology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
HOLDUPS AND EFFICIENCY
WITH SEARCH FRICTIONS
Daron Acemoglu
Robert Shimer
98-14
September 1998
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
fJF1'tnHS\!OlOGY
JAN
2 8 'SSg
Holdups and Efficiency
With Search
Frictions*
Daron Acemoglu
Robert Shimer
MIT
Princeton
December 1997
Current Version: August 1998
First Version:
Abstract
A natural holdup problem arises in a market with search frictions:
make
firms have to
a range of investments before finding their employees and larger investments
translate into higher wages. In particular,
bargaining, the equilibrium
is
always
when wages
inefficient:
are determined by ex post
recognizing that capital intensive
production relations have to pay higher wages, firms reduce their investments. This
can only be prevented by removing all the bargaining power firom the workers, but
this in turn depresses wages below their social product, and creates excessive entry
of firms. In contrast to this benchmark,
we show
that efficiency
firms post wages and workers can direct their search towards
more
is
achieved
when
attractive offers.
This efficiency result generahzes to an environment with imperfect information
where workers only observe a few of the equilibrium wage offers. We show that
the underlying reason for efficiency is not wage posting per se, but the ability of
workers to direct their search towards more capital intensive jobs.
Keywords:
JEL
Efficiency,
Holdup, Search,
Wage
Posting.
Classification: 83
•We thank two anonymous
referees,
Jonathan Levin, and seminar participants
Society for Economic Dynamics Conference.
at
MIT,
NBER
and
Introduction
1
An
investment
Wilhamson
held up
is
if
one party must pay the
(1975) and Grout (1984)
show that incomplete contracts are the underlying
cause of holdups: with complete contracts,
be forced to pay their share of the
cost, while others share in the payoff.
cost.
all
Even
those
who
from an investment can
benefit
in the presence of incomplete contracts,
if
agents arrange their relationships appropriately, holdups are often preventable. Before
making investments, agents can
man and
reallocate property rights (Williamson, 1985, Gross-
Hart, 1986, Hart and Moore, 1990), impose simple breach remedies
and Malcomson, 1993; Edlin and Reichlestein, 1996), or enter
(MacLeod
into long-term relations
(Williamson, 1975).
many
In
situations, however, investments
example, a firm must build a factory before
must complete
must be sunk before agents meet.
can hire workers, and
it
similarly,
For
workers
their education before finding jobs (Acemoglu, 1996, 1997; Davis, 1995;
Masters, 1998). In such cases, contracts and related arrangements are impossible, be-
cause agents do not know
who
their partners will
1996). This suggests that holdup problems
be at the time they invest (Acemoglu,
may be much more
serious in the presence
of trading frictions. This paper analyzes the potential for holdups in markets with
tions,
and examines how markets can
result
is
efficient
internalize the resulting externalities.
ers
Our main
that despite pervasive market incompleteness, the decentralized equilibrium
under
fairly
mild conditions.
It
first
show that when
firms
solves the
is
holdup problem and creates the right
and to participate
incentives for firms to invest in capital
We
fric-
in the production process.
make ex ante investments
before matching with work-
and wages are determined by ex post bargaining, the equilibrium
is
always
inefficient.
Either wages increase with output, creating a holdup problem for firms' investments, or
all
the bargaining power
sive entr}' of firms.
We
is
vested in the firm, leading to very low
is efficient.
who show
Our
levels
and exces-
then turn to an economy in which firms post wages and workers
direct their search towards different firms.
rium
wage
results therefore
We
establish that in this case the equilib-
extend those of
Moen
(1997) and Shinier (1996),
that wage posting can achieve efficiency in the standard search environment.
In these models, where firms do not have ex ante investment decisions, efficiency only
demands that the economy
and Hosios (1990)
equilibrium
efficiency,
for
results as
bargaining power.
number
of jobs.
find that even in the standard search
is efficient
and Shimer's
creates the right
Diamond
(1982)
and bargaining model, the
an appropriate bargaining solution, one can interpret Moen's
showing that wage posting picks the
With ex ante
and so our
Since
efficient distribution of
investments, however, no bargaining solution achieves
efficiency result
is
much more
striking.
Also surprising are our results regarding the role of information: we find that
full
information, whereby workers observe
In particular,
result.
random
When
firms.
it is
wage
all
sufficient for
offers, is
not necessary for this efficiency
each worker to observe the wage offers of two
each firm knows that every worker
who
also observed at least one other wage, there will effectively
among
Finally,
We
and
firms,
has observed
its
wage has
be Bertrand competition
this competition ensures efficiency.
we examine way ex ante wage
conclude that the key feature
is
offers are so effective in
achieving efficiency.
the ability of workers to direct their search towards
firms with different levels of capital, not towards those offering higher wages per
example,
if
firms can
commit
se.
For
to pay a certain share of the output to their employees
(perhaps by committing to play a particular bargaining game), but workers do not
observe the ex ante capital choices, efficiency
that increases
commit
its
capital stock
is still
to different sharing rules
their search, the equilibrium
This
not achieved.
always
efficient.
receive the full benefit of
This finding
is
it
its
investment.
attracts
from attracting more workers, and thus
filling
when
Nevertheless,
more unemployed workers,
The
since
it
is
Wage
ex ante
since workers
increase in profits
vacancies faster, can offset the reduction
Moreover, when firms can commit to a bargaining
wages from investments are exacthj
therefore avoided.
firms
firm that invests
share, they choose a sharing rule which ensures that the decline in profits
is
if
useful in understanding
recognize that a larger investment translates into higher wages.
due to higher wages.
because a firm
and workers can observe investments before directing
investments are observed by workers,
in profits
is
subject to the holdup problem. In contrast,
when workers have bargaining power, a
the essence of our results:
more does not
is
is
offset
due to
liigher
by faster job creation. The holdup problem
posting soh'cs the holdup problem via the
same channel,
formally equivalent to a setup where firms post sharing rules and workers
observe investments before their application decisions. In essence, our economy therefore
achieves efficiency because
it
encourages workers to direct their search towards more
capital intensive firms and enables firms to
The plan
frictions.
is
as follows.
to the appropriate sharing rule.
Section 2 describes the nature of the trading
Section 4 derives the
Section 3 derives the constrained efficient allocation.
equilibrium
holdup
of the paper
commit
when wages
inefficiency.
are determined by ex post bargaining
Section 5 proves that
when
and demonstrates the
firms post wages in order to attract
workers, holdups and search externalities are internalized, even
limited information about the available wages.
efficiency result.
details.
if
workers have very
Section 6 explores the origins of this
Section 7 concludes, and the Appendix contains proofs and technical
The Environment
2
There
is
a continuum
1
of risk-neutral workers and a larger continuum of risk-neutral
firms. All agents live forever in continuous time,
rate
Firms are inactive
r.
allows
them
until they
buy some
and discount the future
capital
>
/c
at the
unemployed worker by posting a vacancy. Holdups
to attempt to hire an
because firms must invest in capital before meeting a worker, and workers
some
of the benefits from larger investments. If a firm employs one worker
/
is
it
and concave, and
satisfies
in
s,
of
Finally,
which case the worker
Unemployed workers
inactive.
reap
and k units
the usual Inada conditions.
each piece of capital breaks down with flow probability
becomes unemployed and the firm becomes
may
We assume that
produces a flow of output f{k), with a price normalized to one.
strictly increasing
which
at marginal cost p,
arise
capital,
common
receive a flow
payoff' of zero.
Matching
is
frictional.
Suppose
in
some labor market there are
Q
G
[0,
oo]
unem-
ployed workers seeking each vacancy. Wlien workers do not direct their search towards
any particular group of
firms,
bor market tightness. This
is
Q
is
for
the (market) queue length, or the inverse of the
example the case analyzed
la-
in the standard bargaining
models of Diamond (1982) and Mortensen and Pissarides (1994). Otherwise, different
vacancies
may be
associated with different queue lengtlas. Matching frictions are modeled
using a standard constant returns to scale matching technology. Each worker matches
with a firm with flow probability ^{Q), and each vacancy matches with a worker with
flow probability rj{Q)
map
and
7]
and
id{oo)
=
and we have
Qfj.{Q),
fj.'
the extended positive real numbers
=
77(0)
=
0.
In words,
if
<
[0,
and
oo]
there are very few
?]'
onto
>
0.
We
itself,
also
so //(O)
assume that
=
77(00)
=
many unemployed
workers per vacancy.
We
These technical conditions guarantee the existence of
locations.
A match
agreement of both
3
An
The
is
consummated
00
unemployed workers per vacancy,
workers find jobs arbitrarily quickly, and firms cannot hire workers; and conversely
there are
/x
also
assume that
interior equilibria
77
and
is
if
concave.^
efficient al-
—turned into an employment relation— upon the
parties.
Efficient Allocation
allocation
is
(constrained) efficient
if it
maximizes the net output of the economy
subject to search and informational restrictions, the standard definition in this literature.
In the Appendix,
we use optimal
the market queue length
Q
and
control theory to solve rigorously for the time path of
capital investment level k that ma:ximizes the value of
'This last assumption is extremely weak. Since ii{q) = ri{q)/q is decreasing,
with the assumption that 77(0) = 0, this is almost a statement of concavity.
ri'{q)
<
ri{q)/q.
Together
we provide a more
net output. Here
intuitive derivation of the efficient allocation.
Let A denote the shadow flow value of an unemployed worker in steady state, so that
an additional unemployed worker
raises steady state output
by A/r. Then, a recm-sion
defining A can be written as:
-A
,^v ff{k)
Since workers are homogeneous,
lations.
it is
destruction,
match ends.
s,
capital until
it
Solving
(1) for
s)pk
matches into employment
hired at the flow rate
is
Accounting
for
by definition
+ s). From
and the
A,
yielding a flow
both impatience,
f{k)/{r
is
fJ.{Q),
is
re-
and match
r,
this,
we must
cost of using k units of
unemployed, the economy
Since each vacancy uses k units of capital, the flow cost
for her.
of maintaining these vacancies
as an allocation that
all
In addition, while the worker
breaks, pk.
l/Q vacancies
is
+
(r
turn
the present value of this gross output
net out the flow labor cost, which
sustains
efficient to
Thus each unemployed worker
of output f{k) until the
A
{r
+ s)pk/Q. An efficient
A,
the shadow value of unemployed workers.
is
maximizes
allocation can
now be
defined
A yields a convenient characterization of the efficient allocation:
Proposition 1 An
{k^,Q^) € (0,00)"
efficient steady state allocation exists.
(i.e.
characterized by a pair
It is
an interior solution) solving
r^{Q)f{k)-{r
+ s + rj{Q)){r +
s)pk
Proof: Appendix
The maximization problem
(2) is
not jointly concave in k and Q, and so the
first
order
conditions are not sufficient for a maximization. Nevertheless, because the efficient allocation
is
an
interior solution to the
maximization problem, the
first
order conditions are
necessary, so they will be useful in recognizing inefficient allocations. These conditions
are given in the following corollary.
Corollary 1 The
efficient steady state allocation {k^
r
^
Equation
r
+s+
+
s
ri{Q^)
+
Q^)
when the vacancy
The
first
satisfies:
(3)
'
s
vJQ') - Q'v'JQ')
+ r]{QS) + {l-QS)Tj'{QS)
(3) is easily interpreted.
value of a dollar
r
,
fraction
is first filled.
impatience and the possibility that the vacancy
on the
This value
may be
f{k')/k'
r
+
right
is
(4)
s
hand
side
is
the current
discounted because of both
destroyed before
it is filled.
The
second fraction
hand
is
the discounted marginal product of the job
if it is filled.
So the
right
side represents the present marginal value of a vacancy using k units of capital.
That must be equal
to the marginal cost of capital
p
at
an optimum. Equation
(4) in
turn requires that the cost of creating another vacancy equals the expected revenue.
Namely,
it
equates the cost of opening one more vacancy pk^ to the additional social
The expression
value from this vacancy, the right hand side of (4) (times k^).
social value
somewhat complicated,
is
since
it
for this
takes into account the reduction in the
matching probabilities of other vacancies.
Wage Bargaining
4
This section examines the search environment of Diamond (1982), Pissarides (1990), and
Mortensen and Pissarides (1994). The preferences, production and search technologies
are as specified in Section
2.
economy
In contrast to Section 3, the
and wages are determined by bargaining between workers and
made
its
investment,
fC
result,
must anticipate how the bargained wage
it
contractual incompleteness
Let
As a
investment and contacted the worker.
is
will
firms, after the firm has
when the
made
in equilibrium,
when
first,
entering the mai'ket earn zero profits,
if it is
in the
mutual
J^ {k) —
interest of the
p/c
=
if
A:
k G
if
G
/C;
result of bargaining
conditional on k since wages
This
An equilibrium
by the wage equation,
may depend on
make a
/C;
third,
profit
second, firms
matches are
worker and firm; and fourth, wages are
determined by bilateral bargaining between employed workers and
summarize the
capital.
its
and J^ {k) denote
firms enter the market, they
maximizing capital investment, so k maximizes J^{k') — pk'
accepted only
makes
the source of holdups in the bargaining model.
denote the set of capital investments
satisfy four conditions:
firm
depend on
the expected present value of a firm with a vacancy and k units of capital.
must
decentralized,
is
w =
firms.
iu{k).
To
begin,
we
This equation
is
the size of the firm's irreversible investment.
Later we discuss the specification of the bargaining
game
in
more
detail.
Analysis
We start by writing the Bellman equations which determine the profit of firms in different
states. Since the focus of this
paper
rJ^{k)
is
=
with steady states, we suppress time dependence.^
f{k)
- w{k) -
sJ^{k):
(5)
'^At this point we assume that workers accept any match, as occurs in the efficient allocation. This
assumption is not restrictive, because the goal of this section is to show that the equilibrium of the
standard search model is always inefficient. If in some equihbrium, some matches are not turned into
emploj'ment, the equilibriimi is necessarily inefficient. In the next subsection, we verify that with Nash
Bargaining, all matches are accepted in equilibrium.
J^{k)
the asset value of a
is
filled
pays a wage w{k), and gets destroyed at the rate
/(fc),
rj^'ik)
The
=
r^iQ) [j'^ik)
value of a vacancy with capital k
which happens
rate
vacancy with capital
generates a flow of output
s.
- .f{k)) -
sJ^'ik).
(6)
due to the possibility of generating a match
is
In the meantime, the equipment breaks
at the rate r]{Q).
down
at the
s.
Equations
and
(5)
imply
(6)
r
Again, the
fraction
+s+
filled
+
r
Comparing equations
(3)
and
+
s
(8),
f'jk)
-
w'jk)
r
+
s
.
r]{Q)
JC solves
^
^'
^
If
w
is
strictly increasing, firms anticipate that investing
neighborhood of the
(7), free
efficient capital stock,
more amounts
is
avoided only
if
w
is
constant
k^
entry implies
r
+
+
s
r
ri{Q)
+
s
this with (3) 3qelds a second necessary condition for the equilibrium to
optimal, w{k^)
^
Since workers do not share in the cost of ex ante
investments, this leads to underinvestment. This holdup
Comparing
The second
a vacancy.
a necessary condition for the equilibrium to be optimal
to bargaining to a higher wage.
Again using
fill
This formalizes the notion that efficiency requires a solution to the
0.
holdup problem.
in the
s
job, as a function of the capital stock.
maximization implies that any k E
viQ)
=
+
fraction represents the time required to
first
(7), profit
w'{k^)
f
r]{Q)
the present value of profits for a
is
Using
is
k. It
=
f{k^)
—
k^ f'{k^). Firms must earn the marginal product of capital,
while workers keep the residual.
effectively solves the
be
If,
holdup problem
for
as
example, workers earned a zero wage, which
w^k) =
0,
the return on capital would exceed
the marginal product, attracting excessive entry. At the root of the excessive entry result
is
the fact that firms create a negative externality
for other firms to find workers.
when they enter,
as they
make
very low, the positive externality vanishes, so entry
down
w{k^)
is
excessive. In
If
wages are
summary, optimality
the level and slope of the wage function in a neighborhood of the efficient
level of capital
=
harder
At the same time, they create a positive externality on
workers because they increase the probability that workers find employment.
pins
it
f{k^)
k^ namely, to achieve
efficiency in the bargaining equilibrium,
-
=
,
k^f'ik^) and w'{k^)
0.
6
we need
Bargaining
Game
We show that these conditions
by the Nash bargaining
we prove the same
that for
axe never satisfied simultaneously
assumption
solution, the usual
if
wages are determined
in this literature. In
Appendix B,
any "regular" bargaining game. Nash Bargaining implies
result for
all k,
P {J^{k) -
=
J'^ik))
-
(1
/?)
- J^)
(J^(A;)
(10)
,
where P is the bargaining power of the worker, J^ is the value of an unemployed worker,
and J^{k) is the value of an employed worker in a job with capital k, defined by
=
rj'^ik)
w{k)
+s[j" -
Her flow value equals her wage minus the expected
which occurs
at the
exogenous rate
we
(l-/3)(r.n(LA
r
Since the production function /
— w
is
bargaining equilibrium
-
from job destruction,
J^{k), and (11) for J^{k)
A;^ is
s
+
{l
r/(Q))
- J^ and
,
r
(3), a
^
-P)V[Q)
concave in the firm's investment
is
concave as
+
+5+
Then
well.
level k, this implies
the investment level in the
(8) implies
the unique solution to
Jl-PMQ)
Comparing with
loss she suff'ers
obtain:
f(lA
net revenue f
(11)
.
s.
Solving equations (5) and (6) for J^{k)
simplifying (10),
J^{k))
+
s
+
-
[l
f'jk)
p)r]{Q)
r
+
^
2)
^'
s
necessary condition for efficiency
is
^
P—
0,
so the firm has
^
the
all
bargaining power.
Now we
can calculate the value of an unemployed worker.
rj^ =/i(Q) [J^(/c^)- J^J.
The
value of an unemployed worker
is
(13)
equal to the flow probability that she finds a
match, times the net present value gain from employment. This equation uses the fact
that in equilibrium,
all
make the same investment
firms
Then
accepted by workers as stated above.
""^
^
/?(,
'
Plugging this into
(9) yields
r
+
and so
/c^,
matches are
(10) implies that
+,+
^(Q))/(fcB)
+ p^.{Q) + {l-PUQy
s
all
^
^
the other necessary and sufficient conditions for a bargaining
equilibrium:
(1
r
-
PMQ)
+ s + p^{Q) +
{l-p)r]{Q)
This allows us to characterize an equilibrium.
f{k^)/k^
r
+
5
_
^
(15)
Proposition 2 A steady
{k^,Q^) e (0,oo)~
(3
is
>
state search-bargaining equilibrium is
and
that solves (12)
An
(15).
If the elasticity of the production function
0.
summarized by a pair
equilibrium exists
is
JAJ
if
and only
if
nonincreasing, the equilibrium
unique.
An
equilibrium never coincides with the efficient allocation {Q^,k^). In particular,
1-
If0<l3<
2.
If
That
^ ."ly^i
<
<
<
/3
either firms
is,
,
either
Q^ > Q^
and k^ > k^ or
1,
either
Q^ > Q^
or k^
^^frisP
<
Q^ < Q^ and k^ <
k^
k^
undermvest (k" < k^ ) or entry
is
too low
(Q
> Q
or both.
),
Proof: Appendix.
This proposition extends Hosios' (1990) results,
choice, the equilibriuna
to the elasticity of the
Then
equation
case.
optimal
(4),
if
and only
matching function,
Suppose capital k
Condition".
models.
is
(3
the efficient allocation
is
= ^(qT)- We
in the
if
and only
if
tions (3)
and
the Hosios condition holds.
(12)."^
Therefore, even though
the social shadow value of labor,
slope of the
standard search and bargaining
characterized by the queue length
capital investment, this bargaining share leads to
it
wage function are equal
is
it is
equal
is
refer to this as the "Hosios
and the equilibrium allocation has queue length
Q^ = Q^
that without a capital
the worker's bargaining share
if
exogenous as
is
who showed
Q^
Q^
satisfying
given by (15). In this
However, with endogenous
holdup problems, as shown by equa-
possible for the level of wages to equal
impossible to ensure that both the level and the
to the appropriate social values.
In light of Hosios' results, search environments are often viewed as quite "neoclassical":
with the right choice of institutional structure to determine the bargaining
strengths of labor and capital, the level of wages
and the
efficient allocation
is
achieved.
is
equal to the shadow value of labor,
Even leaving
aside the difficulty of fine-tuning
bargaining strengths, Proposition 2 shows that decentralization
is
impossible
when
there
are ex ante investments. Search frictions prevent ex ante contracting, and so wages are
forced to accomplish two tasks: encourage investment and discourage entry.
With ex
post bargaining, these cannot be achieved simultaneously.
''This result
is
related to Corollary 2 in Aceinoglu (1996), which shows that in a two-period
with firm and worker investments, the equilibrium
is
inefficient:
distorts physical capital investments, while a low bargaining share of workers distorts
investments.
Here, there are no
distorts the entry margin,
which
human
is
capital investments, but a low bargaining
unmodeled
in
model
a high bargaining power for workers
Aceinoglu (1996).
human
power
for
capital
workers
Wage
5
Posting
Firms commit to and
This section considers a variant of the standard search model.
post wages before nieeting workers in an effort to attract appUcants.
Montgomery
(1991), Shimer (1996),
Moen
(1997) and
Acemoglu and Shimer (1997b)
have also analyzed such a setup. Following these papers, we assume
have
full
information about posted wages. That
is,
Peters (1991),
they observe
for
all
now
posted wages and
then decide which of these to seek. In this decision, they recognize that
workers
who
are seeking vacancies at
worker applying to this wage
probability
fj.{q).
is
wage
w
to firms offering
w
is
q
=
if
T]{q)
=
the ratio of
q{w), then each
hired (and hence actually receives the wage) with flow
Symmetrically, each firm offering this wage expects that
vacancy at the rate
that workers
it
will
fill
its
qiJ.{q).
We distinguish between the
"market" queue length
Q
and the queue length associated
with a particular wage, q{w). With ex post bargaining as
in Section 4, capital-intensive
jobs yielded higher wages (equation (14)), but workers' application decisions did not
respond to these incentives. This was either because workers could not observe firms'
capital choices before
As a
their search.
making
their applications, or because they
were unable to direct
had a common
result in the bargaining model, all jobs necessarily
(market) queue length Q. In contrast, this section allows workers to adjust their application decisions in response to
wage
differentials, so different
with different queue lengths qiw). More precisely,
if
wages are generally associated
firms offer different
wages
librium, then queue lengths will adjust so that workers are indifferent about
in equi-
which wage
to seek. Higher wages will be associated with longer queues.
Search frictions are often interpreted as representing the time required to learn about
a job opening. Since
we assume here
they require a different interpretation.
One
geographic or industrial "labor markets"
labor market.
If
know about
that workers
.
possibility
is
all
of the available wages,
that firms locate in different
Workers know the wage associated with each
they attempt to get a job in a labor market offering a wage of w,
they recognize that there will be on average q{w) other workers competing for each
job opening. Matching frictions exist within individual labor markets, however, so the
worker
is
(1991),
Montgomery
mixed
hired with probability fi{q{w)).
strategies in
An
alternative interpretation follows Peters
(1991) and Burdett, Shi, and Wright (1997). Workers use identical
making
their applications,
and the mixing probabilities are such
that they are indifferent about where to apply (so that mixed strategies are optimal).
If
the realization of the mixed strategy
the worker
is
workers only
is
that there are
employed with probability l/{n
if
at least
+
1).
one worker actually applies
n other applicants
Conversely, firms
for the job,
manage
for its job. In this case,
to hire
the matching
technology corresponds to a standard urn-ball process:
An
equilibrium of the wage posting
game must
=
ri{q)
qfj,{q) oc 1
—
satisfy four conditions:
new
investments and wage commitments are profit maximizing; second,
e"'."*
first,
firms'
entrants earn
zero profits; third, workers direct their search towards the wage(s) that maximize(s) their
expected wealth; and fourth, q{w)
any decision node. More
equilibrium, q{w)
is
consistent with rational expectations beginning at
is
precisely, for
wages
w 6 W,
the set of wages offered- in
i.e.
the ratio of unemployed workers seeking that wage to firms posting
and by the third requirement of equilibrium, applying to such a wage
it,
highest possible utility to workers. For any other wage w'
subgame-perfection requirement with a similar
spirit:
if
q{w')
,
w
gives the
pinned down by a
is
a firm offered w', the queue
length would be sufficiently long that workers would not prefer applying to w' instead
of
ti;
€ W. This
final
requirement
to deviate from the prescribed
is
wage
important
for
understanding the incentive of a firm
set.
Analysis
We
once again start by writing the Bellman equations, restricting ourselves to steady
states.
The
value of a vacant firm posting wage
rj^\tu, k)
This
is
=
v{q{w)) (J^{w, k)
identical to equation (6), except that
so that q{io)
^ Q^
it
w
and using capital k
- f'{w,
k))
- sJ^\w,
is
k).
allows queue lengths to
(16)
depend on wages,
Similarly.
rJ^[w., k)
=
which has exactly the same reasoning
f{k)
-w- sJ^{w, k),
(17)
as (5) in the previous section.
For workers, the value of being employed at wage
w
is
rj^{w)=iu + s{f' -J'^iw)).
(18)
Note that the value of an employed worker only depends on her wage, not the
investment
k.
firm's
In fact, workers do not need to observe firms' investment decisions. In the
bargaining equilibrium of Section
4,
the value of employment depended on k, because
"This matching technology does not satisfy our requirement
corner solution in the social planner's problem,
i.e.,
an
also lead to nonexistence of a bargaining equilibrium.
7;(oo)
=
/t(0)
=
oo.
This
may
lead to a
with no active firms. It may
does not alter the main conclusion
efficient allocation
However,
it
wage posting equilibrium is efficient.
^We once again impose that workers accept any match. This is not a restriction since our definition
of equilibrium ensures that q{iu) =
if workers prefer unemployment to employment at a wage of w.
So any wage offered in equilibrium will be accepted by workers.
in this section, that the
10
the bargaining solution split the surplus in the match.
In contrast, here the
wage
is
determined solely by the firm's ex ante decision.
Next, the value of an unemployed worker applying to a job with wage
w
is
rJ^{w)=^l{q{w)){J'^{w)-J").
(19)
Again, the construction of these equations parallels (11) and (13), with the value of an
unemployed worker defined by the highest value that she can attain while unemployed:
=
J"
sup J^{w),
(20)
toew
W
where
is
the set of wages offered in equilibrium.
we formalize the requirement that
Finally,
pair of conditions which apply for
all
w
w, including
=
q{w)
(i)
q{-) satisfies rational
expectations as a
W:
^
a J^ > J^ w]
Q
(21)
(ii)
where
wage
J^' is
The
first
J'^iw)
condition ensures that workers do not apply for a
(even off the equilibrium path), unless
ic
unemployment J^
value of
q{'w)
defined by (20).
J^ >
—
for all
w <
.
it
Manipulating (18) and
'
''"'"^^Iq)
rj^.
them
gives
The second
utility at least
(19), this
is
equal to the
equivalent to requiring
condition ensures that unemployed
workers can never expect to earn more than the value of unemployment.
they determine q{w) for
An
all
equilibrium of this
w, and in particular ensure that
J'^
economy can now be defined more
and
J^
distribution of wages
J^\w,
argmaxu,,/;
0.
It is
k)
and capital investments
—pk, and satisfying the
X
nonempty
a
maximizing
for all
w >
succinctly as the appro-
priate Bellman equations, J^{w,k), J^{w,k), J^{w), J^{w),
a queue length function q that satisfies (21);
= J^ {w)
Together,
as described above;
joint support, of the
firms' profits,
free entry condition,
i.e.
maxw^k -/^(^, k)
X C
—pk —
simpler to characterize an equilibrium as the solution to a constrained optimiza-
tion problem:
Lemma
librium
1
if
(g,
X)
with {w^, k^) G
and only
if
X
and
w,k,q
^^^^
r
+ s + yL[q)
r
+
is
a steady state wage posting equi-
,
,
w
(22)
,
-—
subject to
''In
q{w)
[w^, k^, q^) solves:
max
fected
=
q^
s
-\-
r][q)
r
+
>
pk.
s
imposing these conditions, we are implicitly assuming that the value of unemployment is unafa single firm's decision. Tliis is natural in our atomless economy, and is formally equivalent
b}-
to the limit of a finite agent
economy (Burdett,
Shi,
11
and Wright,
1997).
Any
wage, capital, and queue combination observed in equilibrium must mciximize the
utility of
the representative worker, subject to
moglu and Shimer (1997b) prove
new
vacancies earning zero profits. Ace-
Lemma, and we do
this
another triple {w',k',q') gives workers more
Intuitively, if
not repeat the proof here.
utility,
and
entry condition, then a firm could offer a slightly lower wage than w',
and make
ers,
Lemma
in this
attract work-
make the same investment and
complete information environment:
not degenerate, there must exist two
yield firms the
wage posting game. Gener-
unique solution to this constrained optimization problem, and so generi-
ically there is a
cally all firms
still
the free
strictly positive profits.
allows us to characterize an equilibrium of the
1
satisfies
same
More important
profit
(g'^,
//
the same wage. This result
diff'erent
k^)
is
intuitive
wage, investment, and queue triples that
and workers the same
and
is
the wage and/or capital distributions are
if
utility.
for the focus of this paper, the equilibrium
coincide. Intuitively, (2)
Proposition 3
ofi"er
and
efficient allocations
(22) are equivalent optimization problems. Formally:
an
efficient allocation as characterized in Proposition 1,
and
w' = f{k')-k'f'{k'),
then there
Conversely,
is
wage posting
a
if (g, A')
then {q'^,k^)
is
an
is
equilibrium.
(<?,
A")
(23)
with q^
=
a wage posting equilibrium with q^
q{uj^)
=
and {w^,k^) 6 X.
q{w) and {w^,k^) G X,
efficient allocation as characterized in Proposition
1.
Proof: Appendix.
Equation (23) shows that there
wages and productivity.
Thus the absence
an increasing equilibrium relationship between
is
Firms that undertake larger investments pay higher wages.
of a holdup problem
(i.e.
the fact that the equilibrium
appears to contradict the intuition from the bargaining equilibrium. The
that firms here are not compelled to offer higher wages
instead can conceive of investing
more while keeping
when they
their
is
efficient)
diff"erence is
invest more, but
wage constant. They
offer
higher wages precisely becaiise they want to attract more workers.
So at the margin,
the higher wage that a capital intensive firm offers
by the faster rate
of job creation.
In fact, the
wage
is
is
exactly
off^set
always equal to the marginal product of capital
(equation (23)), because at a given wage, firms adjust their capital investment until this
condition obtains.
The second
part of the efficiency result
is
that entry decisions are optimal.
from the definition of the optimal allocation that jobs
(or labor markets)
We know
with capital
investment k^ and queue lengths q^ produce more net output than any other possible
combination. Since in any equilibrium,
all profits
12
are driven to zero, the expected present
value of wages must be higher in jobs offering k^ and q^ than in any alternative. So
these jobs are offered,
all
if
workers will be drawn to these jobs giving them the highest
expected wages. Therefore, no other allocation can be an equilibrium and there are no
profitable deviations from the efficient allocation with k^
result
that wage posting induces unfettered competition
is
shows, this ensures that in equilibrium worker utility
is
and
q^.
among
The
essence of this
and as
firms,
Lemma
1
meiximized.
Imperfect Information
The assumption
know about
that unemployed workers
and fortunately not necessary
for
most of our
all
the available wages
Proposition
unchanged.
is
The
rest
not affect the efficient equilibrium described in
will
we summarize the main
Here,
3.
This
strong,
Suppose that each worker only
results.
observes two posted wages, independently chosen from the set of vacancies.''
of the setup
is
result of this subsection:
Proposition 4 Suppose each worker only observes two independently drawn wages from
wage
the
k^
distribution.
Then, there exists an equilibrium in which
attract queue length
,
Q^ and
offer
wage
w^ =
all
firms choose capital
— k^ f'{k^)-
f{k^)
Proof: Appendix.
The
intuition for this result can be seen as follows:
starting from the equilibrium
iq^ ,k^ ,uj^\ a reduction in workers' information reduces the profitability of
viations, but does not raise the profitability of
,
realizes that
=
this
is
offers
j^f
it
) s\
will
w^
only attract workers by giving
incompatible with the firm making positive
a wage w'
<
them the
all
de-
other firms
different
wage
efficient level of utility
However, we know from the perfect information benchmark that
w^,
it
will
profits.
More
specifically, if
a firm
obtain a shorter queue length, as described by the perfect
information indifference condition (21).
And
if it
longer queue length — but only up to a point.
(i.e.
Because
wage w^ with associated queue q^ any firm that posts a
are offering
J^
any others.
some
the market queue length), only
2Q
If
offers
a wage w'
> w^
,
it
will
obtain a
the unemployment- vacancy ratio
is
Q
workers observe a given firm's wage, so a deviat-
ing firm's queue length cannot exceed SQ. As a result, the second part of condition (21)
may be
violated for sufficiently high wages.
game reduces
This change in the extensive form of the
the profitabihty of some deviations from prescribed equilibrium, without
raising the profitability of any others. Therefore,
the
it
does not change the fact that there
This analysis borrows from our earlier paper Acemoglu and Shimer (1997a), where we endogenized
amount of information gathering by workers, but did not investigate the efficiency issues discussed
here.
13
is
no profitable deviation when
all
firms
make investment k^ and
off^er
the wage w^.
This remains an equilibrium.**
By
reducing the profitability of some deviations, however, we
In particular, take a local
equilibria.
lem
{w*,k*,q'').
(22),
If
there
is
maximum
may
introduce other
of the constrained optimization prob-
<
no {w,k,q) with q
to this problem, thon this will be an equilibrium.
2q''
This
is
that yields a higher value
unlikely to be a real issue,
however, as simulations suggest that the constrained optimization problem has a unique
maximum, [w^ ,q^ ,k^),
local
the efficient allocation. Moreover, as the
observations of each worker increases, inefficient equilibria
become
number
of
wage
progressively less
likely.
Another way of obtain the intuition of
this result
them
firm has one opponent, but price competition forces
The
efficient allocation is similar, since firms
utility subject to non-negative profits. Diff'erently
know who
the firm does not
its rival is.
by comparing our imperfect
With Bertrand competition, each
information environment to Bertrand competition.
cost.
is
to set price equal to marginal
choose wages to maximize workers'
from standard Bertrand competition,
This does not matter, however, because
it
knows
that the rival also maximizes workers' utility subject to zero profits.
The
result in this subsection
is
also related to Burdett
and Judd (1983), who show
in
a model of price search that when each consumer observes two random prices, Bertrand
competition
is
obtained.
However, our result
is
stronger since in Burdett and Judd's
paper, firms have infinite capacity, so each firm knows that the other firm observed by
the consumer can supply her with the rccjuired good.
In contrast, in our setting each
firm can only hire one worker, so the otlier firm whose
wage
may
not hire the worker. Nevertheless,
firm
is
sufficient to take us to
is
observed by the worker
expectation of competition with this "other"
tlic
Bertrand competition.
Understanding Efficiency
6
With the
while
traditional bargaining setup, the equilibrium
when
(Section 5).
is
always inefficient (Section
firms post wages and workers direct their search, efficiency
It is
important to understand what underlies these
results.
is
4),
guaranteed
We can
begin by
noting that there are two important diff"erences between the environments of Sections 4
and
5:
in Section 5, firms are able to post
Although there
'^This result
is
wages and workers can direct their search.
a natural association between these features,
it is
possible to inquire
would change substantially if each worker only observed the wage of one firm. Then
its wage slightly would not suffer a reduction in queue length, since workers have
It is easy to show that the unique equilibrium of such a model has firms paying a zero
a firm that lowers
no alternative.
(monopoly) wage. This
is
the
Diamond
(1971) paradox.
14
which one
At
is
the source of the strong efficiency results we obtained in Section
a trivial level, the possibility of directing search
commit
important.
is
to wages, but workers applied to firms randomly, firms
and we would obtain the equilibrium
to a zero wage,
of Section 4 with
three hybrids of the environments analyzed in Sections 4 and
1.
Wages are determined by ex post Nash
5.
The fundamental
equilibrium
is
=
0.
To
we consider
5.
and
direct their search appropriately, as in
condition of Section 5 which determines applications
workers must have the
decisions will once again be an equilibrium condition:
same expected
seriously,
/3
bargaining, as in Section 4, but workers are
able to observe firms' capital investment
Section
firms could
If
would obviously commit
and wage posting more
investigate the role of directed search
5.
utility at all jobs (with positive
queue length).
As a
result,
the
characterized by the constrained optimization problem (22), with
one additional constraint: wages are not a choice variable, but instead are set
ex post by Nash bargaining.
So
the additional constraint (14).
in this hybrid environment,
It is
does not bind at the equilibrium
straightforward to see that constraint (14)
(efficient)
values of k^
and
5
and only
if
p
equal to the elasticity of the matching function
rj.
is
and
(i.e.
wages must satisfy
at the efficient allocation) if
,
w^ and
q^ of Sections 3
workers' bargaining power
In other words,
Hosios "bargaining power equals elasticity" condition holds,
we obtain the
allocation as the equilibrium of this hybrid environment. Therefore,
can direct their search towards firms with more
if
2.
the Hosios condition
Firms commit
to
and advertise a bargaining
they direct their search accordingly.
there are
when workers
capital,
holdups are avoided
rule
before the
if
only
rule,
(3
matching
stage.
but not its capital investment,
The equilibrium
and
coincides with the bargain-
where Q^ is the market queue length. As in
P =
,gV)
now holdup problems. More specifically, capital investments
ing equilibrium, with
4,
efficient
is satisfied.
Workers observe each firm's bargaining
Section
when the
,
are given by (12), so there
is
underinvestment. Firms compete efficiently along the
dimension that workers can observe, which ensures the Hosios condition. Lairger
investments translate into higher wages but do not attract a longer queue as workers
do not observe investments, so there
rewarded
3.
is
a holdup inefficiency: &:ms are not fully
for their investments.
Firms can commit
to
and advertise a bargaining
rule
P
Workers can observe each firm 's bargaining rule and
direct their search accordingly.
The
equilibria of this
15
before the
matching
stage.
capital investment, 'and they
economy coincide with the
efficient equilibria cliaracterized in Sections 3
the capital investment
These hybrid models
is
and
5.
clarify the role of different ingredients of
and
Hosios condition solves the holdup problem.
The reason
some
bill is
choice of
/3
conditional
on
formally equivalent to a wage commitment.
workers" observe the capital choices of firms
take
A
our economy.
When
direct their search accordingly, the
is
that, even
though wo'kers
of the retm'ns from investments in the form of higher wages, the higher
wage
exactly offset by the increase in profits due to the longer queues that are attracted.
Holdups do not arise because workers' application decisions internalize the dependence of
wages on investments. Although the assumption that workers observe
perfectly direct their search
is
extreme,
it
all
wages and can
seems plausible that workers have some idea
about prevailing wages and can direct their search to some degree. They can choose,
for
example, between jobs in the manufacturing and service sectors. The forces emphasized
paper should therefore reduce the scope of holdups in practice.
in this
Complementing the
the Hosios condition
that
of
all
is
which solves the holdup problem when
role of directed search,
satisfied,
wage
or bargaining rule
commitments (posting) ensure
firms offer a division of output satisfying the Hosios condition.
wage commitments and directed search therefore
Note conversely that holdup problems
depends on
its
capital investment,
and
The combination
leads to an efficient allocation.
arise if the value of a firm's
this investment
is
wage commitment
unobserved. This
is
consider-
ably weaker than a more naive conjecture that holdups arise whenever wages increase
with investment. Remarkably,
ered,
the natural search environment that we have consid-
in
whenever workers can direct
known
their search
capital intensity, the Hosios condition
between two randomly chosen jobs with
is all
we need
for
workers to internalize the
dependence of wages on investments. In particular, when the Hosios condition holds,
a firm that chooses a larger investment pays higher wages but also attracts sufficiently
longer queue lengths so that
tives.
We
its
investment incentives are aligned with
do not have a very good intuition
for
why
exactly the Hosios condition
required to balance the two opposing effects on firm profits, although a
understanding can be obtained by thinking of each investment
land"
economy and workers' job applications
We know
from Hosios (1990) that when the
efficient incen-
more
level as a
is
intuitive
separate
"is-
as decisions to enter one of these islands.
elasticity condition
and expected wages are ma:x;imized within each
is
satisfied, net
output
island, so workers will choose to enter
the island with the highest expected wages, which will be the one with the greatest net
output, so efficiency will be achieved.
To conclude, with wage posting and
both
in the entry
directed search, the market achieves efficiency
and investment margins, despite the large number of missing markets
and possible widespread imperfect information about available wages. This
16
is
because
the wage posting environment of Section 5 enables two distinct but related phenomena.
First,
allows workers to direct their search towards
it
more
capital intensive firms.
Even
environment of Section 5 workers do not care about firms' investment de-
though
in the
cisions,
they observe wages, and firms that
make
larger investments offer higher wages.
Second, wage posting ensures that wages satisfy the Hosios condition, the other require-
ment
of efficiency. This
is
Shimer (1996) discussed
Moen
related to but stronger than the results of
in the introduction, since
it
(1997) and
obtains in a model with ex ante
investments, where bargaining equilibria are always inefficient.
Concluding Comments
7
A
natural conjecture
holdups and
it
to this conjectmre.
in
when
there are ex ante investments and trading frictions,
we formalized
is
In fact, our
not always true.
main
result
With standard assumptions on the form
is
this conjecture,
in striking contrast
of trading frictions,
an
which workers can direct their search towards more capital intensive jobs
achieves efficiency.
and workers
The
that
inefficiencies are unavoidable. In this paper,
and showed why
economy
is
We
showed that
direct their search towards higher
results presented here
may
with: perhaps with a sufficient
inefficiencies.
this result arises naturally
We
wage
when
firms post wages,
firms.
suggest the opposite conjecture to the one
commitment technology, trading
believe that this conjectm'e
would
also
frictions
we
started
do not lead to
be incorrect, because there are
other, harder to avoid, inefficiencies, once again arising from frictional trading
and the
informational problems underlying search. For example:
1.
W^e have assumed that there
is
only one-sided investment.
The
inefficiencies
em-
phasized in Acemoglu (1996) and Masters (1998) rely on two-sided investments.
It is
unclear whether the ability of firms to post complex contracts can prevent
inefficiencies in this
2.
An
type of environment. This
is left
for future
important cause of inefficiencies in search models
is
work.
random matching: with
two-sided heterogeneity, the equilibrium matching configurations
with those preferred by a social planner. This problem
may be
may
not coincide
avoided
if
workers
have complete information about the available jobs, but that assumption seems
too strong in an economy with an atomless distribution of jobs.
With imperfect
information, skilled workers will sometimes have to match with low capital firms,
and mismatch
will
be present. In
firms' capital investments,
in
this case, the skill distribution of
and via
this channel, also
wages
(as in
workers affects
Acemoglu, 1996;
the context of bargaining). Therefore, the return to worker and firm investments
will
depend on the actions of other firms and workers, introducing
17
externalities.
3.
We
treated the information structure as exogenous.
how much
here)
and
to search
also
both in the sense of determining reservation wages
how much
that workers have
an informational
In practice, workers decide
is
information to gather.
When
endogenous, the equiUbrium
externality. In particular, workers
will
(as
the amount of information
be
inefficient,
because of
do not take into account the
impact of their information on the wage distribution, and thus on the
firms
and the wages of other workers
(see
18
we have
profits of
Acemoglu and Shimer, 1997a).
Appendix
A
Proofs of Main Results
Proof of Proposition
prove existence.
A
dependence.
We
characterize
first
and then
tlae efficient allocation,
are looking for a steady state solution, and therefore suppress time
"social planner" chooses the
firms' capital investments k,
output. That
We
1:
time path of the market queue length Q,
and the unemployment rate u to maximize the value of net
is:
maxf
(.«).(0(())(^-pM<)) -3|(r +
.«0)e-*^
(24)
subject to
u{t)
The
constraint (25)
rate:
is
=
s{l-u{t))-fx{Q{t))u{t).
a standard equation describing the evolution of the UBemployment
the increase in unemployment
hires.
(24)
is
a bit
risk neutrality)
equal to the flow into unemployment minus
is
for the interest rate (equal to workers' rate of
and depreciation. The
from newly created
The number
jobs.
times the probability that each
is
first
of
(r
+
is
time preference under
in parentheses represents the payoff
the
is
hired, u{t)fi{Q{t)).
f{k{t)) units of output until the capital
must continue to rent
term
new jobs
new
firms renting capital at a cost {r + s)p,
more complex. Think of vacant
which accounts both
(25)
number
A
of
unemployed workers
newly created job produces
destroyed. However, in the process, the job
s)pk{t) units of capital. Therefore, the net expected present
value of a newly created job
is
of maintaining open vacancies,
,['}''])
i.e.
—
pk{t).
The second term
represents the cost
the rental cost of a vacancy, times the
by each vacancy
unfilled vacancies u{t)/Q{t), times the capital used
k{t).
nmnber
of
All payoffs
are discounted back to an initial time.
The maximization
of (24) subject to (25) gives the planner
not afforded by the model. She
may
costlessly adjust the capital investment of existing
vacancies after they are created, but before they are
would never take advantage of
problem described here
is
this option.
filled.
Thus the
In steady state, the planner
solution to the maximization
the solution to the social planner's problem.
restrict the planner to choose the
follows
an extra degree of freedom
same investment
from concavity of the objective
for all jobs.
in k for arbitrary
More
subtly,
The optimality
we
of this
Q.
Write the current valued Hamiltonian associated with this dynamic optimization
problem.
//(/c,
g,
uA) = u L{Q)
U^^
- pk\ 19
^1±^\
+
A
(5(1
-
tx)
-
^.{Q)u)
Let {k{X),Q{X)} maximize H{k,Q,u,\) for a given value of A.
independent of u
>
since
0,
u
Crucially these are
enters the maximization problem Uneaxly:
{k{X),Q{X)} e argrnaxMQ)
fc,Q
' >)
(^-pf^
\r
+
s
Define the Maximized Current Value Hamiltonian
n{u,X)
=
^^^.
Q
'
J
(26)
as:
H{k{X),QiX),u,X).
Arrow's generalization of Mangasarian's Sufficiency Theorem (Kamien and Schwartz,
1991, p. 222) states that the following conditions are necessary and sufficient for
to be a steady state solution to the
function of u (in fact,
rX
and u
—
=
satisfies
it is
dynamic optimization problem:
=
a linear function), k
=
+
fi{Q)
k{X),
Q=
a steady state version of the state equation
s(l
Solving (27) for A
is
Q, u}
a concave
Q{X),
—— -pkj -
I
?i
{A;,
X[s
+
(27)
i^[Q)),
(25):
-u) =^(Q)u.
(28)
:
^^v{Q)i^-{r + s + viQ))pk
{r
+ s)Q +
Substituting this into (26) and simplifying,
maximization problem
Now we
lem.
Q
is
(2).
[0,
oo]^,
is
=
0,
since f{k)
the Inada conditions.
zero
when Q =
Finally,
=
0.
It is
It is
negative
negative
oo, since /i(oo)
=
we prove that there exist
when
maximization prob-
continuous in
when
Q
is
k
is
The
{k,
Q) defined on
value of the objective
sufficiently large, since
sufficiently small, since r]{0)
zero
^^ ^ by
— 0. And it
Q that yield positive payoff, implying
interior value. Fix Q with ij^Q) positive and
Then by the fundamental theorem
s)p,
which
-{r +
+ v{Q)){r + s)p
dK >
{r + s)Q + T]{Q)
I
'
v{Q)f'{k)-{r + s + v{Q)){r + s)p
dK =
{r + s)Q + T]{Q)
L
s
20
exists
by the Inada
of calculus, the value of the objective at
{k,Q)\s
r' r?(Q)/'(Ac)
is
values of k and
maximum must be obtained at an
Choose k >
to satisfy f'{k) = ^^^^jj^{r +
conditions.
this
0.
that the
finite.
to the static
implying existence of a maximum. Next, extremal values of k
yield zero or negative value to the objective (2).
/c
Thus the solution
the efficient capital and queue length.
observe that the maximization problem
First,
when
is
the text
we obtain
can establish the existence of an interior solution to
the compact set
or
(1) in
v{Q)
where the inequahty exploits concavity of
>
/: /'(^)
f'{k) for all k
<
k,
and the equality
uses the definition of k.
Proof of Proposition
2:
Existence: Consider the graph of equation (12)
rant of the plane. Since
By
the Inada conditions on /,
=
K(Q). Moreover, K(0)
+
by f'{ki)/{r
implicitly
and /
increasing
is
rj
=
concave, the graph
is
upward
sloping.
implicitly defines a continuously differentiable function
it
=
since 7^(0)
s)
is
ik,Q) space, the nonnegative quad-
in
and
=
/'(O)
oo;
and K(oo)
=
ki,
defined
p.
Next, divide (12) by (15) and simplify to obtain:
M^)
+s+
r
Given the assumptions on
mapping
Q(0) >
(0,
00) onto
77,
and
Thus
itself.
Note that
if /3
=
the
is
=
hand
side
Q
unique
<
a decreasing function of Q,
=
(0, 0).
an equilibrium, since we have assumed that k G (0,oo). More
equilibrium capital stock sequence converges to zero too,
is
not allowed, this equilibrium sequence
Uniqueness:
If
the elasticity of /
Q
relationship between
and
and hence
(29),
Then
{r
+
either rj{Q^)
<
or k^
>
since
A;^,
Next, consider
r
The
right
/9
>
s){r
side
is
^
0,
precisely, as
k^ ->
0.
/9
But since
V
+
S
+
-
the
=
nonincreasing, (29) describes a nonincreasing
most one intersection between (12)
at
0.
(3)
+s+
and
(12) imply:
rj
is
<
/?
is
p<
T]{Q^))
{r
<
f'{k^).
increasing and /
<
^^Q^"
(1
(^)
+
- Q^)v'{Q^) ~
(^^)
r
+s+
7]{Q^))'
Q^ < Q^
This implies that either
concave.
is
^^^
s){r
This
is
the desired result for
™P^y-
+ s + {l-0) 7?(Q^) + /3/.(Q^)
decreasing in p. Therefore, substituting for
with
Jqb\
implies
jviQ')
A;
not
not lower hemi-continuous.
is
Thus there can be
k.
t]{Q^) or f'{k^)
+ s + TjiQS) +
hand
is
i.e.
However, this
most one equilibrium.
at
Efficiency: Take any
and
two curves cross at
00, the
an equilibrium, establishing existence.
the cm-ves intersect only at (Q, k)
0,
is
solves this equation, Q(/c). Since
Since Q(A;i)
0.
.29)
^^^^
1
kf'ik)
left
for all k, a
Such an intersection
ki.
_ZR _
.
PHQ)
/?,
above (12) when k
0, (29) lies
some k <
/i,
-
{1
Q'ri'jQ'))
r]{Q^)
+
(1
nk')/k'
- Q^v'jQ^)) f{k^)/k^
+ t]{Q^) + (1 - Q^)7]'(Q^)
{vJQ^)
- Q^WiQS) ^
21
r
+
s
'
<P
Hence
> f{k^)/k B
either f{k^)/k^
- Q'v'iQ')
^
+ v{Q^) + (1 - Q^WiQ^)
viQ^) -
viQ')
+
r
s
r
+ +
s
QV(Q^)
+ (1 - Q^WiQ^)
r?(QS)
<
Since /'satisfies the Inada conditions, the former possibility imphes k^
Alterna-
k^'.
tively,
d
v{Q)-Qn'{Q)
(
dQ Vr +
5
where the inequality
Combining the
Q^ > Q^
,
+
7?(Q)
results for
>
/9
and
Thus the
77.
<
(3
constraint. Eliminating
(^, q)
.qVn
3:
implies that either
w
At a solution, the constraint
w
and
must lead
is
binding. Otherwise,
without violating the
from the objective through the binding constraint yields
if
they are
(2).
efficient.
equation, observe that the objective in (22) does not depend on k;
only the constraint does. Since the constraint
in k
in (22)
raise the value of the objective,
pair appear in equilibrium only
To obtain the wage
change
Q^ > Q^
k^ > k^ and
latter possibility implies
or both inequahties are reversed.
would be possible to reduce
Thus a
^-v"iQ){ir + s)Q + r]{Q))>0.
(l-Q)7j'(Q)
exploits concavity of
Proof of Proposition
it
+
binding, optimality dictates that a small
is
to a violation of the constraint, or at least keep the constraint
binding. Equivalently,
V{q){f{k)-w)
r
+
s) {r
+
s
+
-pk] =
0.
r]{q))
Simphfying the derivative with the binding constraint yields
Proof of Proposition
4:
To formalize the
(23).
issues related to imperfect information,
we introduce some additional notation. Let p{w, w') be the
who observes wages w and w' applies to w in preference of
and w' are mutually
p{w,w')
=
1
exclusive, p{w,'w')
whenever J^{w) >
decisions of other workers.
J^ =
where
G
is
utility of
1.
Since applying for
w
Applications are optimal
if
w'
.
J^{vj'), implicitly taking into account the application
The expected
value of an unemployed worker
the equilibrium wage distribution.
The term
in parentheses
an unemployed worker conditional on observing two wages
Finally,
we need
all
to define q{vj).
indifference across wages.
In-
is:
(p{w,w')J^'{w)+p{w\w)j"{w'))dG{w)dG{w'),
j j
integrals take expectations over
anteed.
+ p{w\w) =
probability that a worker
possible realizations of these
With
full
information,
it
w
is
the maximal
and w' and the
,
two observations.
was pinned down to ensure
With imperfect information, such
indifference
any subgame, a worker might not observe some wages, and
22
(30)
is
not guar-
for this
reason
cannot apply to them, even though she would
Instead, the equilibrium
like to.
queue
function satisfies
q{w)
Q
where
is
the equiUbrium
is
2Q
(31)
•
number
unemployed workers per
of
firm.
The equilibrium
w
is
equal to the number of workers
since there are
Q
workers per firm and each observes two wages, times the
queue length
which
= 2Q Jp{w, w')dG{w'),
wage
for a
who
observe this wage,
wage instead of the alternative
probability that each of those will apply for this
w',
a
wage randomly drawn from G.
An
equilibrium
is
once again defined as the appropriate Bellman equations; a queue
length function q satisfying (31); a joint support of the wage and capital distribution A!,
each element of which maximizes firms' profits and
satisfies
wage distribution G; and a preference function p that
p{iu, w')
=
whenever J^{w) > J^{w').
1
Now we
the free entry condition; a
satisfies
the optiraality condition
can give the proof of Proposition
Proposition 3 implies that {w^ k^ Q^) solves (22).
,
,
It trivially solves
imization problem with the additional constraint q € [0,2Q^].
maximized value
of this program,
4.
the same max-
Letting J^* be the
{w^ k^ Q^) must similarly solve the dual program
,
,
V{q){f{k)-w)
T—- - pk
max
w,k,q
(r
subject to
r
and
+
s) (r
+ s + ri[q))
Z'^'",
+ s + fi{q),
> J";
qe[0,2Q^].
We prove that this dual program is in fact the problem that a firm faces when it considers
deviating from a proposed equilibrium with
First, profit
maximization and
wage w^
is
will also observe a
it
The
J^*.
wage
wage of w^
in expectation,
Thus the value
deviating firm realizes that
.
promises them at least J^*.
its
it
Thus
On
it
B
solution
is
In Section
4,
wage w^
all
workers
the other hand, since only
this
is
who
observe
2Q^ workers
its
its profits,
this.
will
wage,
observe
Thus the firm
subject to these two constraints.
an equilibrium.
With Alternative Bargaining Rules
we showed that no equilibrium with Nash bargaining achieves
Here we extend that
k^ and
to a worker of applying for a job
cannot have a queue length longer than
{w^ ,k^ ,Q^),
Inefficiency
oflfering
be imable to attract any applicants unless
will
attempts to maximize the expected value of
As the
other firms
free entry drive the other firms to use capital
drive the market queue length to Q^.
at
all
result to arbitrary bargaining rules that satisfy
ditions. For this purpose, consider
weak
efficiency.
regularity con-
a bargaining game with transferable utility between
23
two
players,
there
where conditional on agreement they obtain
2,
disagreement,
is
so agreement
A
and
1
obtains
1
mutually
is
and 2 obtains
rfi
We
beneficial.
^2,
joint surplus y.
where we assume that
+ do <
di
denote the bargaining payoff of player
bargaining rule A, with payoffs uf (y, ^1,^2) and U2
= y — uf
is
regular
if
If
be
i
y
Uj.
the following
conditions are satisfied:
1.
Individual Rationality: Vy
>
di
+ ^2,
wf (y,
2.
Weak
>
y
>
+
Monotonicity:
>
ut{y',di,d2)
The
first
di
d2,
if c?i
>
more than
and u^(y,
<^i
<
uf-{y,di,d2)
di, (i2)
<
y
<
—
y
—
c?2-
d2 then
u^{y,di,d2).
condition states that both players obtain at least as
The second
points.
Vy'
^2)
c?i,
condition requires that
his outside option,
and the
if
at
some
much
as their disagreement
level of surplus
each player receives
level of surplus is increased,
then each payoff
increases.
As examples consider two well-known bargaining
uI
—
—
di
—
uf-
=
di ii
P{y
imposes
some P G
We
all
di;
u^
/?
=
y
G
—
[0, 1].
d2
if
Nash bargaining imposes that
Rubinstein-Shaked and Sutton bargaining
Py >
y
—
do]
and uf
= Py
otherwise, for
straightforward to verify that both surplus divisions satisfy both
values of
G
/?
[0, 1].
next state:
Proposition 5
is
some
di for
py <
It is
[0, 1].
conditions for
+
(^2)
rules:
If
w{k)
determined by a regular bargaining solution, the equilibrium
is
always inefficient.
Proof: Recall that
Also, di
w'{k^)
=
=
efficiency requires:
= J^
J^{k) and ^2
is
only possible
1.
either w{k)
2.
or J^(fc5)
The second
w{k^) < f{k^)
=
=
in this case.
=
/(/c"'^)
-
k^f'{k^) and w'{k^)
Weak Monotonicity then
=
0.
implies that
if:
rj^.
J^(it^).
condition
-
w{k^)
is
not possible as long as r]{Q)
k^f'{k^), leading to suboptimal entry.
24
<
00.
The
first
implies that
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