MIT LIBRARIES
I
3 9080 02618 0965
•II'
I
!iiiti'.!!titiHS}ir;Btitii)i
J«415
no;07-13
2007
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/persistentapprecOOcaba
'DEWEY
HB31
.M415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Persistent Appreciations
and Overshooting:
A Normative Analysis
Ricardo
"^^
J.
Caballero
Guido Lorenzoni
Working Paper 07-13
April 19,2007
RoomE52-251
50 Memorial Drive
Cambridge,
02142
MA
This
paper can be downloaded without charge from the
Network Paper Collection at
Social Science Research
httsp://ssrn.com/abstract=98 1 909
and Overshooting:
Persistent Appreciations
A Normative
nicardo
J.
Analysis
Caballero and Guido Lorenzoni*
MIT
and
NBER
April 19, 2007
Abstract
Most economies experience episodes
question arises whether there
paper we present a model of
justified
is
if
the export sector
is
of persistent real excliange rate appreciations, wlien the
a need for intervention to protect the export sector.
In this
destruction where exchange rate intervention
may be
irre\'ersible
is
financially constrained.
However the
not whether there are bankruptcies or not, but whether these can cause a large exchange
rate overshooting once the factors behind the appreciation subside.
ex-ante and ex-post interventions. Ex-ante
effects
this case the
bulk of the intervention takes place ex-post, and
if
On
is
policy includes
the financial resources in the export sector are relatively abundant.
of the depreciation phase. In contrast,
policy
The optimal
during the appreciation phase) interventions
(i.e.,
have limited
in
criterion for intervention
if
shifted toward ex-ante intervention
the methodological front,
is
concentrated in the
the financial constraint in the export sector
and
it is
first
is
In
period
tight, the
optimal to lean against the appreciation.
we develop a framework
to study optimal
dynamic interventions
economies with financially constrained agents.
JEL Codes:
Keywords:
EO, E2, FO, F4, H2.
Appreciations, overshooting, financial frictions, irreversible investment, pecuniary
externality, real wages, optimal policy, exports.
'Respectively, caball@mit.edu and glorenzo@mit.edu.
We
are grateful to Arvind Krishnamurthy, Pablo Kurlat,
Enrique Mendoza, Klaus Schmidt-Hebbel and seminar participants at Berkeley, Harvard, IMF, MIT,
WEL-MIT,
their
the World Bank, San Francisco Fed,
LACEA-Central Bank
comments, and to Nicolas Arregui, Jose Tessada, Lucia Tian, and,
assistance.
Caballero thanks the
NSF
for financial support.
of Chile,
especially,
First draft:
IFM-NBER,
and Paris School of Economics
Pablo Kurlat
September 2006.
for
for excellent research
This paper circulated
previously as "Persistent Appreciations, Overshooting, and Optimal Exchange Rate Interventions."
^
Introduction
1
Most economies experience episodes
of large real exchange rate appreciations.
For example, they can stem from domestic
factors with the potential to fuel these appreciations.
policies
aimed
taming a stubborn inflationary episode, from the absorption of large capital inflows
at
caused by domestic and external factors, from exchange rate interventions
from domestic consumption booms, from a sharp
economies
so-called
terms of trade in commodity producing
disease).
While there are idiosyncrasies
when the
rise in
in trading partners,
most extreme form, from the discovery of large natural resources wealth (the
or, in its
Dutch
There are many
appreciation
common
in each of these instances, the
policy element
persistent enough, the question arises whether there
is
is
that,
is
a need for inter-
vention to protect the export sector (often referred as "competitiveness" policies). This widespread
concern goes beyond the purely distributional aspects associated to real appreciations. The fear
that
somehow
If this
is
concern
is
justified,
More
too late?
medium and
the
In this paper
dynamic model
long run health of the economy
compromised by these episodes.
should policymakers intervene and stabilize the exchange rate before
generally,
what does the optimal policy look
we propose a framework
of entry
is
and
to address this
exit in the export sector
and exchange rate stabilization may be
justified. In
the export sector's ability to recover, the
cwiimon policy element.
where entrepreneurs face
our model,
when
economy experiences a
We
illustrates three recent examples:
mid
of the early 1990s,
where the appreciation
overshoots
its
1990s,
is
and
The
present a
financial constraints
financial constraints
damage
large exchange rate overshooting
when
not always present, overshooting episodes are pervasive, especially
1
Finnish,
Although
financial frictions are
Mexican and Asian episodes
In each of them, the pattern
late 1990s, respectively.
it
like?
once the factors behind the appreciation subside and nontradable demand contracts.
widespread. Figure
is
one
is
followed by a depreciation of the real exchange rate that significantly
new medium term
level.
In our model, the overshooting results from the export sector's inability to absorb the resources
from the contraction
(labor) freed
in real wages,
'The
figure
which
shows
the Asian Crisis,
is
real
in
nontradable demand. This inability leads to an amplified
costly to consumer- workers.^
There
is
scope for policy intervention because
exchange rate indices normalized to lOO'at date
November 1994
for
Mexico, and October 1991
fall
0.
for Finland.
The
latter
The Asian
corresponds to June 1997
crisis real
exchange rate
for
is
a
simple average of the indices for Indonesia, Korea, Malaysia, Philippines, and Thailand. Data Source: IPS (Efi'ective
Real Exchange Rate) for Pinland, Indonesia, Malaysia and Philippines. Real Exchange Rate from
Hausmann
et al
(2006) for Mexico, Korea and Thailand.
^In practice, the drop in the relative price of nontradables and real wages often takes the form of a sharp nominal
depreciation which
is
not matched by a
Valdes (1999) and Burnstein et
al
rise in
(2005).
the nominal price of nontradables and wages. See,
e.g.,
Goldfajn and
t-24 t-21
1-18
t-15
1-12
t-9
t-6
1-3
t+1
1+4
t+7
t^lO
t+U
1+16 t-19 t-22 t+25 t+28 t+31 t+34 t+37 t+40 t+43 1+46
Months
-Asian
Figure
1:
Crisis (avg)
1990's Overshooting in Finland, Mexico
and Asia
there
a connection between the severity of the overshooting and the extent of the contraction in the
is
export sector during the preceding appreciation phase.
for
nontradables in this phase, then there would be
consumers were to reduce their demand
If
less destruction
ex-ante and a faster recovery
ex-post. However, rational atomistic consumers ignore the effect of their individual decisions during
the appreciation phase on the extent of the overshooting during the depreciation phase.
pecuniary externality that
Our
and informs policy intervention
The former
of factor reallocation in the presence of financial constraints.
by transiting into an appreciation phase, which
starts
in our framework.
two parts, a positive one and a normative one.
analysis has
dynamic model
justifies
There are several regions
Poisson probability.
When
first
consists of a
Our model economy
phase with a
exits into a depreciation
by the financial resources of
of interest, indexed
the export sector at the onset of the appreciation.
economy reaches the
it
It is this
financial resources are plentiful, the
best as real exchange rates (and real wages) are pinned
down by purely
technological free entry and exit conditions, and hence are orthogonal to consumers' actions.
lower levels of financial resources, financial constraints
phase, the depreciation phase, or both.
If
may become
binding during the appreciation
they are only binding during the appreciation phase,
then the economy experiences bankruptcies but the recovery of the export sector
the depreciation phase starts and the exchange rate
factors. In contrast,
if
of the export sector
is
On
to
the financial constraint
slow and the
the norrriative side,
maximize consumers'
we
At
is
initial real
is
again pinned
down by
is
swift once
purely technological
binding during the depreciation phase, the recovery
depreciation overshoots the long run depreciation.
consider the optimal intervention of a benevolent planner that seeks
welfare, subject to not worsening entrepreneur' welfare.
The planner
faces
the same financial constraint present in the private economy. This limits the planner's instruments,
as
it
rules out direct transfers across groups.
Instead,
implemented through interventions that influence the
affect
real
we
focus on market-mediated transfers
exchange
rate.
That
consumers' choices and, thus, the entrepreneurial sector through their
prices.
Prom
whenever
this perspective,
we show that consumers gain from
this leads to a faster recovery of the
is,
interventions that
effect
on equilibrium
stabilizing the appreciation
export sector once the appreciation subsides.
The
gain derives from the increase in real wages associated to a faster reconstruction of the export
sector.
Importantly, even
when overshooting
siderations put limits on
how much
is
expected, intertemporal consumption allocation con-
intervention
is
desirable during the appreciation phase. This
connects our analysis to a central consideration for policymakers
ciation,
If
there
be
is
should be
it
when
dealing with an asset appre-
the currency, real estate or any other asset with potential macroeconomic implications:
a need for intervention,
left for after
how much should be done
as prevention (ex-ante)
and how much
"the crash" (ex-post)? In our framework, the answer to this question de-
pends primarily on the extent of the financial constraint
in the
export sector.
On
one end, when
the financial constraint
the constraint
is
severe, ex-ante intervention
is
loose, ex-post intervention
is
most
most desirable and
is
On
effective.
when
the other end,
In general, the optimal
effective.
policy has elements of both, ex-ante and ex-post intervention.
The approach
to optimal policy proposed in this paper resembles that of the literature
dynamic optimal taxation. In
this dimension, the
main innovation
methodology to an environment where a subset of agents are
on the
restrictions
ability of policy to reallocate resources
the economy. This approach and the solution
of the paper
on
to apply this
is
financially constrained, imposing
between these agents and the
method developed should prove
rest of
useful outside our
particular application.
Our paper belongs
to an extensive literature
on consumption and investment booms
economies, as well as on the role of financial factors in generating inefficiencies in these
e.g.
Gourinchas
et al 2001,
Caballero and Krishnamurthy 2001, Aghion et
growing theoretical and empirical literature on the
exporters
constrained export firms
in innovation
There
(see,
is
a
on the behavior of
The
is
of this
idea that excessive exchange rate fluctuations can hurt financially
also present in
Aghion
et al (2006)
,
who
explore
its effects
on investment
and growth.
The pecuniary
externality that justifies intervention in our framework
is
related to those identi-
Geanakoplos and Polemarchakis (1996), Caballero and Krishnamurthy (2001, 2004), Loren-
zoni (2006) and Farhi et
to
booms
and Manova, 2007). Our paper emphasizes the general equilibrium implications
behavior (overshooting).
fied in
effect of financial frictions
open
Chaney, 2005, Manova, 2006), and on the slow response of exports after devaluations
(e.g.
(Fitzgerald
al 2004).
in
embed
al
(2006). Aside from
this externality in a tractable
model
its specific
context, the
of optimal policy,
main novelty
which allows us to
of our paper
is
fully characterize
the economy's dynamics and to analyze the trade-off between ex ante and ex post interventions.
In terms of
intervention
doing
is
its
mechanism, the paper
justified
(see, e.g.
by the presence
of
also belongs to the hterature
dynamic technological
on Dutch
externalities
disease.
There,
through learning-by-
van Wijnbergen 1984, Corden 1984, and Krugman 1987). In contrast, our paper
highlights financial frictions
and the pecuniary
externalities that
stem from these.
The
policy
implications of these two approaches are different: While learning-by-doing offers a justification for
industrial policies as a development strategy, the financial frictions
we
highlight have intertemporal
reallocation implications of the sort that matter for business cycle policies.
Section 2 presents a stylized model of creative destruction over appreciation and depreciation
cycles. Section 3 characterizes
different extensions
and
their
optimal exchange rate intervention in such setup. Section 4 discusses
impact on optimal
extensive appendix, containing
all
the proofs.
policy. Section 5 concludes
and
is
followed by an
A
2
Simple Model of a Destructive Appreciation and Overshoot-
ing
In this section
appreciation.
we present a model
The export
of an
economy experiencing a temporary, but
persistent, real
sector faces large sunk costs of investment, which limit the extent of
its
desired contraction, in order to keep capital operational and preserve the option to produce once
the appreciation
If this
financing
over.
is
is
However,
this waiting strategy generates losses that require financing.
limited, the export sector experiences a larger contraction than desired.
the point of view of the
economy
as a whole, these excessive contractions
recovery of the export sector once the appreciation
real depreciation
is
over, leading to a
From
may compromise
the
prolonged period of deep
and low wages.
2.1
The Environment
There
is
a unit mass of each of two groups of agents within the domestic economy: consumers and
entrepreneurs (exporters). There are two consumption goods: a tradable and a nontradable good.
The consumer
supplies inelastically one unit of labor each period, which can be used as an input for
the production of tradables or nontradables. In both cases one unit of labor
one unit of output.
"export unit,"
i.e.,
is
needed to produce
In addition, the production of tradables requires one unit of capital, or an
the technology of the tradable sector
is
Leontief in labor and capital. Creating
an export unit requires investing / units of tradable goods. After an export unit has been
it
needs to be maintained in operation, otherwise
it is
set up,
irreversibly shut down."^ Entrepreneurs are
the only agents that have access to the technology to run and maintain export units. At date
they begin with n_i open export units.
The markets
for tradables,
nontradables and labor are
competitive.
Entrepreneurs are risk neutral and only consume tradable goods. Their preferences are given
by the
utility function
oo
where
c^
''^
>
denotes entrepreneurs' consumption of tradable goods.
Consumers have
separable instantaneous utility on the consumption of tradables and nontradables, cf and
Their preferences are given by the
logCf
.
utility function
EJ2P%{u{cr)+u{c^))
t=d
^
These assumptions capture the
fact that export oriented firms often
erations than firms producing primarily for domestic markets.
generalization. Later in the paper
we
have more specific (sunk) capital and op-
Of course there
are important exceptions to this
discuss the effect of introducing adjustment costs in the nontradables sector.
where u
The
(c)
~
and
log c
taste shock
take two values in
is
S=
6t is
the only source of uncertainty and
{A,
D} and
The economy begins with a
period, with probability
with
We
=
6t
a taste shock.
tt
follows a
Markov process with
[D\A)
=
5,
economy switches
the
D
st
=
A, with
n
—
dt
{st+i\st).
state,
i.e.,
vr
Each
9a.
= D,
{A\D) = 0.
to the "depreciation" state
an absorbing
is
which can
st,
transition probabihty
transition into the "appreciation" state
9d- Once the latter transition takes place,
assume
determined by the state
is
st
that:
9a>0d =
1.
In the appreciation state the taste shock drives up consumers'
demand
for
both tradable and
nontradables, putting upward pressure on the real exchange rate (since the supply of tradables
is
fully elastic while that of tradables
demand
not - see below).
is
are usually due to positive wealth shocks,
commodity producing country,
taste shock
is
complications.
e.g.,
In reality, increases in consumption
an improvement
terms of trade for a
or to external shocks which generate positive capital inflows.
a convenient device to introduce a shift in consumption
We
in the
will return to this issue later in the paper,
The
demand with minimal added
once we have developed our main
points.
Both groups have access
state contingent securities.
to the international capital market, where they can trade a
On
pay one unit of tradable good
of securities are denoted
to date
in
t.
by a
each date
in period
(s*)
where
+
s*
1 if
=
state St+i
(so,5i,
Note that our simple Markov chain yields
•••,
Sj)
is
realized.
The entrepreneurs
holdings
denotes the history of the economy up
histories that are limited to a block of periods
A, followed by D's (there are no alternations).
All entrepreneurs begin with an initial financial positions equal to gq.
it
of
agents trade one-period state-contingent securities that
t,
t
full set
to zero without loss of generality.
Consumers
For consumers, we set
face no financial constraints, while entrepreneurs
face the financial constraint
a{s*)
That
is,
entrepreneurs cannot commit to
>0.
make any
(1)
positive
repayment
at future dates.
This
is
a simple form of financial markets imperfection, which captures the idea that entrepreneurs have
limited access to external finance. This
The
rest of the
world
is
is
the
07ily friction
t,
in the
model.
captured by a representative consumer with. a large endowment of
tradable goods and linear preferences represented by
neutral: at date
we introduce
E^^q/3*Cj
'*.
Therefore asset pricing
the price of a security paying one unit of tradable in state sj+i
is /^tt
is
risk
(st+i|s().
,
Decisions and Equilibrium
2.2
Let p
(s*)
denote the price of the nontradable good
in
terms of units of tradable (the numeraire), or
the real exchange rate (defined a la IMF). Given the linear technology in the nontradable sector, the
equilibrium wage in terms of tradables must be equal to this price. Consumers and entrepreneurs
take the real exchange rate as given. Equilibrium prices and quantities are functions of the whole
history
t,
s*.
notation, whenever confusion
To save on
shorthand
e.g., pt is
for
is
not possible we only use the time subindex
(s*).
p
Consumers
2.2.1
Since wages are equal to pt, markets are complete, and intertemporal prices are pinned
down by
the world capital market, consumers face the single intertemporal budget constraint
Y: P'^
where
tt
(s*)
(^*)
(c^ {s')
+P
{s')
c^
{s'))
denotes the ex ante probability of history
< Y.P'n
s'.
{s')
[s')
p
(2)
,
Thus, consumers' demand
for tradables
and nontradables take the simple form:
d
=
—
Pt
where the constant «
is
There are two important features from the consumption block.
demand curve
is
for nontradables shifts
endogenous and
is
upward. This
is
First,
during
A
the source of the appreciation.
increasing in the value of the exchange rate at any future date.
periods the
Second, k
The
latter
feature will be essential in the analysis of optimal policy.
2.2.2
Exporters and Equilibrium
Even though consumption
both firms
and consumers,
this issue in detail,
It is useful to
volatility is not the result of
if
we need
any
may
friction, it
create problems for
the export sector has limited financial resources.
Before discussing
to understand exporters' decisions.
separate the entrepreneurs' decisions regarding consumption and investment from
the problem of creating
sector that creates
new
units.
To
this end,
we assume that
and destroys export units and makes zero
there
profits.
is
a competitive adjustment
Let
qt
denote the price of an
export unit. Equilibrium in the adjustment sector requires that
That
is, if
qt
e
[0,/],
qt
=
/
qt
=
(4)
if
nt
>ni_i,
(5)
if
ni
<
(6)
nt-i.
must be equal
units are being created their price
to the creation cost /, while
they are
if
being destroyed their price must be zero.
The
entrepreneur's flow of funds constraint can then written as
c^^-[s')+q{s'){n{s')~n[s'''))-\-P
<
<St+i\st)a [{s\ st^^,))
Y.
{l
-p
[s'))
n
{s')
+a
{s')
,
(7)
Each period, the entrepreneur uses
new export
finance consumption, investment in
Notice that our timing assumption
productive,
i.e.,
is
Let
a
V
and investment
units,
his financial wealth, at, to
in state contingent securities.
that production units created at date
they immediately generate unitary profits of
The entrepreneur chooses sequences
utility,
—pt)nt, and
his current profits, (1
for
c^''^
(s*),
n
1
are immediately
— pt.
and a
(s'),
t
(s*)
to
maximize
his
expected
subject to the flow of funds constraint (7) and the financial constraint (1) for each history
(a (s') ,n {s*~^)
(s*) units of
;
s*)
cash and n
denote the expected
(s'"'')
F(a(s%7^(s*-l);s')=
utility of
an entrepreneur in state
s*
who
is
s*.
holding
production units. Then we can write the Bellman equation:
max
c^'^ (s*)+/3
J2
(st+i|sf)
vr
^
(a ((sS Sj+i)) ,n (s')
;
(sSsj+i))
s.t.
(7),
a
It is
c^'" (s*)
>
0,
st+i))
>
Ofors(+i€S'.
{{s'-
,
n
(s*)
>
0,
straightforward to setup the entrepreneur's problem in sequential form and argue directly that
the value function
is
linear in a (s')
and n
(s*~^), given that
constraints are hnear.^ Moreover, a (s^) and
(7), in
the form a
V
where
4>
(s*)
''For the
finite.
We
(s*)
+
{a {s')
q (s')
,
n
n
(s'~^) only appear in the flow of funds constraint
that the value function takes the form
(s*~-^). It follows
{s'-')
;
s*)
=^
(.*)
+
<^
{s')
•
(a [s')
+q
(.*)
n
{s'^'))
,
(8)
represents the marginal retm^n on entrepreneurs' wealth.
moment, we suppose that the
will
n
both the objective function and the
prices {p (s') ,q (s')} are such that the entrepreneur's expected utility
check later, case by case, that this condition
8
is
satisfied in equilibrium.
is
}
In each period, the entrepreneur chooses
how many production
consume, and how many contingent claims to purchase. The
mentary slackness conditions) with respect
-{q{s')-{l-p{s')))cp{s')+l3
-0(s*)
The
first
-
T^{st+i\st)<f>{{s\st+i))q{{s\st+i))<0, n{s')>0,
J2
>
0,
c^'^ (s*)
+ 0((s*,Si+i»
<
0,
a((s',s,+i))
>0,
forallst+i
losses
if
Remember
pt-
e5.
(11)
condition states that the opportunity cost of the resources used in keeping the marginal
a unit in operation corresponds to the price of acquiring that unit,
-
(9)
(10)
0,
The
unit in operation must be equal to the expected value of that unit tomorrow.
1
to
to these choice variables are^
<
(^
how much
order conditions (and comple-
first
(s*)
1
units to operate,
and these
that an open unit must remain active, so
losses
pt
minus the current
qt,
>
the firm
1
is
profits,
making current
add to the cost of keeping the unit open. The second condition states that
the entrepreneur's consumption
Otherwise,
if
cost of keeping
is
must be equal
positive, the marginal value of wealth
to one.
can exceed one. The third condition says that the marginal value of wealth must be
it
non-increasing between two consecutive histories. Holdings of financial assets can only be positive
between two
Finally,
histories
we
ties
[c^
(s*)
,
are optimal;
is
constant.
are in a position to define a competitive equilibrium.
A
Definition 1
where the marginal value of wealth
competitive equilibrium
c'^ (s*)
(ii)
,
c-^'^ (s*)
,
n
(s*)
,
a
is
given by a sequence of prices {p
such that:
(s*) }
(i)
,
q (s*) }
and quanti-
the consumer's decisions [cF (s*)
the entrepreneur's decisions [c^''^ [s^) ,n [s^) ,a (^s^^^
quences [n (s*)} and {q (s*)} satisfy (4)-(6);
(s*)
(iv) the labor
,
c^
are optimal; (Hi) the
market clears for each
(s*)
se-.
s*,
n(s*)+c^(5')=l.
2.3
The Appreciation and Depreciation Phases
Recall that our
The
economy
starts with a stock of export units, n_i,
situation that concerns us
is
one in which there
is
and creation during the depreciation. Moreover, we
option to wait
is
sufficiently positive that
it is
and has
just entered state A.
destruction of units during the appreciation,
also wish to focus
not optimal to destroy
all
on a scenario where the
export units during the
appreciation.
^The envelope theorem implies that the Lagrange
multiplier on the budget constraint
is
equal to
(p
(s'
An
2.3.1
The export
Efficient
Benchmark
a benchmark,
us
let
study a case where ao
first
never binding. In this benchmark, there
current state
st,
A
D
and
in the
A
using
we
since, as
will see,
it
sufficiently large that financial constraints are
switches to D.
n
in the
A
is
(p
(s*) is
we
will
simply index variables
and
D
when
destruction
Correspondingly, g^
constant and equal to one in both phases.
=
the
and qd
phases, respectively, reduce
fully
economy
=
f
enters phase A, and creation
It
follows that the first order
0,
-f + {l-VD)+Pf -
0,
determine the real exchange rate in each phase:
W,
p^"
=
p^^
= 1-(1
+
1
(12)
-/?)/.
(13)
assume that
(l-/3)/<l,
ensuring that creation
is
profitable in the
Given these prices we can
where
k-^^ is
equal
find the
TJb
_
TJb
_
D
Market clearing
phase and that p^^
>
0.
fbn
NJb
_
fb
NJb
_
K'
^
in
each state:
^A
^^''
1-(1-^)/'
to:
yields the
number
fb
^By "A phase" we mean
=
(Al)
consumption of tradables and nontradables
l-P{l-6)
2{{1 -(3)9 A + SI3)-
fb
s*
We
to:
-
{\-pA) + 5Pf
We
except for the
or D.
conditions for
which
s*
equilibrium prices and quantities are constant both in the
phase.^ Therefore, with a slight abuse of notation,
later that in equilibrium there
when
is
no need to keep track of the history
is
In the absence of financial constraints,
show
As
sector has financial resources ao to finance the losses during the appreciation phase.
all
1
of units
f^
open
in
^A
each state:
'"
fb
the histories of the form
{A,...,A,D,...,D).
10
s'
1
=
r-i
{A,...,
A).
By "D phase"
all
A
those of the form
1,5
P
1
0.5
Figure
now easy
It is
to see that the following
the economy enters state
from
A
A
at date 0,
First Best
2:
two assumptions guarantee that there
and that there
>
1
1
>
when
when
economy
shifts
the
W
W
(A2)
+
(A3)
l-(l-/3)/'
Notice that, as long as the preference shock 9a
is
sufficiently large, the equilibrium prices are
determined on the supply side of the model, by (12) and
the appreciation
when
Figure
is
such that the current
the switch to the
2
D
losses,
During the
I,
during
equal the opportunity cost of creating a
summarizes the benchmark economy, plotting the equilibrium dynamics of the
A
phase,
'The parameters
= Po
pA —
(13). In particular, the price
phase occurs, SPf in expected value.
exchange rate and of the number of export units
p_i
positive creation
destruction
+
1
unit
is
is
to D:
n-i
fully
14
12
10
it
is
to generate this figure are
D
state to the
expected duration when
cS
=
in the event the appreciation lasts five periods.''
optimal for the economy to accommodate the increased demand for
as conventional initial conditions.
transition from the
real
A
j3
=
These
state in period
0.97, 6
initial
0.
We
0.2.
11
=
0.2,
f
=
3,
conditions arise
plot
Sa
if
=
the
2.1.
We
choose n-
Jt-
and
economy makes an unexpected
an appreciation lasting
5 periods, since that
is its
However, since shutting down units
nontradables by contracting the export sector temporarily.
wastes creation costs,
it is
with each of them incurring flow losses of
The
is
>
also optimal for the export sector to keep n;^
—
p)^
units in operation,
1.
The
following Proposition summarizes the case of high entrepreneurial wealth.
cutoff
d-^''
derived explicitly in the Appendix.
Proposition
isfies
A
the
>
ao
and
wealth,
(p
(First best) There
1
a cutoff a^^ such that
is
d^^, then the equilibrium real
D
if the
constant and equal to
'
exchange rate and the number of firms are constant within
The marginal value of entrepreneurial
phases, and are given by (12), (13), and (14).
(5*), is
entrepreneurs initial wealth sat-
1.
The Constrained Economy and Overshooting
2.3.2
Suppose now that ao
not large enough to implement the
is
first
best path
<
ao
(i.e.,
are two margins through which this deficit can materialize. First, the export sector
enough resources
can,
if it
it
may
—
to finance the flow of losses (p^
not have enough resources
left
l)7^}i
d-'"'').
may
There
not have
during the appreciation. Second, even
to finance the investment
/
(
n^j
— n^
j
when
the
appreciation phase ends.
Relative to the
state
St,
case,
we now need
is still
D
a gradual transition in the
is
stationary, with constant prices
D
phase started. The reason
for this
is
that
phase where the export sector rebuilds and
At the same time, due to complete markets the
constrained by limited financial resources.
phase
to keep track not only of the current exogenous
but also of the number of periods since the
in this case there
is
benchmark
and
quantities.
The next proposition summarizes
A
the
properties of equilibrium prices and quantities that will be usefid in the following characterization.
Proposition 2
phase,
in
p
(s*)
,
If ao
n
(s')
D. The price
,
<
d^''
and a
q (s*)
is
then p
(s*)
n
(s*)
,
is
the
number
a (s*) and
4>
(s*)
,
and a
only depend on the
equal to zero in the
This proposition allows us to
where j
,n
(s')
^vrite
p
of periods the
(s*)
A
(s*)
number
phase and
= pA
are constant in the
in the
of periods that the
to
f
in the
A phase
economy has spent
D.
in
D
and p
A
phase.
In the
D
economy has spent
phase.
(s')
= pd,j
in the
Analogous notation
is
D phase,
used for
(s*)-
Let us focus on the
A
phase.
The
relevant optimality conditions for the entrepreneur are
(l-p.4)f4
+
W'/'D,0
=
0,
(15)
and
4>A>
"-0,0
(I^D,o
12
>
0.
(16)
The
A
stationarity of entrepreneurial wealth in
=
implies that a a
ao
and the budget constraint
(7)
can be written as^
-
(1
The
(1
-
=
5)P)ao
(pa
- l)nA +
PSaofi.
(17)
flow generated by the initial resources ao can be used to finance the operational losses of the
export units that remain open during the appreciation, and to transfer financial resources to the
recovery phase in D. Going back to the
(16) distinguish the case
a£)_o
>
0)
aofl
=
0).
The
in the
order conditions, the complementary inequalities in
where the financial resources are used
for
both purposes
{(f)A
—
(Pd o ^^^^
A
(^^
>
4>dq ^iid
and the case where they are only used to cover operational
first
A
first
ha (equation
order condition for
losses in
an expression
(15)) yields
for the real
exchange rate
region:
PA
=
1
l
axA^fl ^ fb
+ l^^f-I—
<Pa^
.
'Pa
where the inequality comes
production units incur
first-best,
from. (16).
p^ >
losses, as
and entrepreneurs are
1.
As
benchmark
in the
When aofl >
indifferent
case, the appreciation
the real exchange rate
is
between holding state contingent
is
such that
equal to that in the
securities or holding
production units. This indifference means that these two assets have the same expected return,
which pins dovra the equilibrium exchange rate as
o-Dfl
=
wedge
0,
is
the expected return on export units
is
unconstrained economy. Instead, when
in the
larger than that
on state contingent
possible because only entrepreneurs can pmrchase export units,
constrained. This depresses the exchange rate to a
pA smaller than p^
and they are
many production
This
financially
Far from being good news,
.
this smaller appreciation reflects the fact that financiallj' constrained firms are
as
securities.
unable to keep open
units as they would like and hence are forced to reduce production
and labor
demand.
For given parameters, we can show that the
initial level of ao
determines which of the two cases
discussed in the previous paragraph arises in equilibrium.
Proposition 3 (Constrained appreciation phase)
the real exchange rate in the
in the
A
phase
Let us focus
side
is
pA < Pa
now on
A
phase
^"'^ '^D.o
=
There
and aofi >
is p^^
is
0,
a cutoff d'^
while if ao
<
<
a^^ such that
d"^ the real
the case where oq
<
d"^.
=
1.
PA
d^''
we have
<p^
>
1
>
d'^
exchange rate
To determine ua, note that from the consumption
UA +
<
a^
0.
and labor market equilibrium, we have
'As long as oo
if
and so ca
is
set to zero (see the
13
proof of Proposition
2).
.
Solving for p^ and substituting into the budget constraint (17) pins
down
the
number
of production
units that are kept active during the appreciation:
ao(l-(l-<5)/3)=f-^^^-l')n^
For a given
k,
lower financial resources ao reduce the
open during the appreciation (the right-hand
side
Notice, however, that in general equilibrium k
extreme
cases,
Let us
an economy with lower ao
now turn
to the
entrepreneurs consume and
in the stationary
D
region.
<^d^j is
is
of production units that are kept
increasing in jia) and lead to
falls as
may end up
well (see the
equal to
argument below
first
we have that
= l-il-P)f = p{,'
benchmark
export sector, after the recovery
1
is
- -^ >
Pd
completed,
level.
1
-
=
^
Pd
the benchmark economy,
demand
also
The
is
depressed and hence the export sector eventually expands to
and the
f{nD,j
-
<pD,j
first
where J
in the
benchmark
case,
riDj^i)
>
1>
order conditions for the transition are:
== (1
<Pdj
>
-PD,j)nD,j,
(18)
^D,j+l^
(19)
{-f + l-PD.j)4>D,j-\-Pf4>D,3 + l
J,
in
not reached instantly since financial constraints also hamper the recovery
flow of funds constraint
...,
follows that the long
consumers are poorer than
absorb the labor freed by the smaller nontradable sector. However, unlike
this stationary state
is
n;^
economy not only entrepreneurs but
is
it
demand
is:
Since in the constrained
0,
completed,
phase of D:
no =
=
is
the equilibrium condition in the labor market and the fact that consumers'
size of the
for J
in 2.3.3) so, in
order conditions,
lower in the constrained than in the benchmark case (as we show below),
phase.
destruction.
with a higher level of n^.
Thus, from the
1.
Eventually, the real exchange rate converges to the
run
more
Starting backwards, once the recovery phase
po,j
From
number
is
=
the last period of the transition phase in
(20)
^,
D
and where no-i
is
equivalent
notation for ua.
Equation
sector.
The
(18) states that during the recovery phase, firms use all their profits to rebuild the
inequalities in (19) reflect the fact that the financial constraint
recovery and gradually declines, and hence there
Reorganizing
(20),
we obtain an expression
PA. =
is
is
on
in the
no reason to accumulate "cash" or to consume.
for the real
exchange rate during the transition:
l-/(l-/3^)<l-/(l-/3)=p{?.
14
tightest early
That
is,
during the recovery phase the depreciation
refer to this
is
deeper
when
the economy
is
We
constrained.
deeper depreciation as the overshooting implication of financial constraints.
The presence
of overshooting
means that wages
are not only lower than in the
benchmark case
during the appreciation phase, but also during the transition phase of D. This observation closes
our argument, as
K
why consumption
explains
it
consumers' lifetime income (see
reflects the
smaller than in the
levels are lower in the constrained case, given that
(3)),
and
by
that, history
history, the
The exchange
.^
with dashes
shooting.
rate appreciates in the
phase, as in the
,
The export
the recovery
A
each panel) and in the depreciation phase
in
is
sector contracts during the
only gradual during the
value of a unit of wealth, which
is
summary
Let us conclude with a
D phase.
D
A
it
A
experiences a large and protracted over-
phase and, unlike in the benchmark economy,
region, drops sharply
D
of the marginal
upon the
transition into
region.
proposition:
the real exchange rate throughout the
exchange rate in the
=
and P-i
benchmark economy (represented
Proposition 4 (Constrained depreciation phase and overshooting) There
> a^
= ho
The bottom panel shows the path
highest in the
D, and gradually declines within the
that if ao
pt are
first best.
Figure 3 depicts the constrained economy, assuming for simplicity that n_i
Pp
wages
phase overshoots
its
D
phase
is
p^
,
is
<
d^^ such
< a^
the real
a cutoff d^
while if a^
long run value early on in the transition.
< Pd for j = 0, 1, ..., J and poj — Pq for j — J +1, for some J > 0. (Note
depending on the model's parameters).
d^ may be greater or smaller than a
PD,j
...
That
is,
that the cutoff
,
General Equilibrium Feedback
2.3.3
Our
discussion above highlights the export firms' problem for a given consumption
ever, firms' actions affect households'
constraint on firms, the lower
lation
between ao and
reaches
its
maximum
Note that
ple,
k.
labor
demand and income. This feedback
Figure 4 plots this relation and shows that k
>
for ao
this general
is
income through labor demand. The tighter
d^'^
(see
Lemma
when
ud =
1
— k/p^ which
rises as
poorer consumers allocates a larger share of
Up
to now,
we have developed a model
section turns to the other
The parameters
are the
main concern
same
its
the financial
captured by the
re-
increasing in ao until
equihbrium feedback generates some counterintuitive
it
results.
For exam-
Even though export firms are more
financially
financial resources are low, in the long
see this, recall that
is
2 in the Appendix).
the model has a sort of sclerosis as ao declines.
constrained
is
is
demand. How-
run they absorb a larger share of
n.
To
k drops. This simply says that an economy with
resources to satisfy foreign than domestic demand.
of equilibrium destruction
in this paper. In particular,
as those used for Figure
2,
15
plus ao
=
0.5.
and overshooting. The next
it
shows that when ao <
d^^,
p
1
2
4
Figure
3:
10
12
14
10
12
14
10
12
14
Constrained Economy
16
J
1
0.44
^^^^
0.43
^^
y^
X
^
0.42
0.41
0.4
-
0.39
-
0.38
i
-
\
t
1
,
1.5
0.5
Figure
the social planner
(and hence
3
may be
4:
Consumers' Income and.Financial Constraint
able to raise k by inducing consumers to choose a different path for
c^
nj).
Optimal Ex-ante and Ex-post Intervention
In the previous section
we showed that when the export
the depreciation phase following a persistent appreciation
rate overshooting (a sharp real
implicitly, in practice
it is
wage
sector has limited financial resources,
may come
with a protracted exchange
decline) while the export sector rebuilds. Either explicitly or
this overshooting
phase that primarily concerns policymakers and leads
to a debate on whether intervention should take place during the appreciation phase. In particular,
the concern
may be
is
exposing
In this section
is
whether by overly stressing the export sector during the appreciation, the economy
itself
to a costly recovery phase once the factors behind the appreciation subside.
we study
this policy
problem and conclude that
indeed scope for policy intervention.
equilibrium
is
not constrained
The reason
efficient, as
for
if
an overshooting
such intervention
consumers ignore the
is
is
expected, there
that the competitive
effect of their individual decisions
on the severity and duration of the overshooting during the depreciation phase.
The optimal
policy includes ex-ante and ex-post interventions.
focus of intervention
is
ex-ante, and the bulk of
it
There are instances when the
consists in stabilizing the
exchange rate during
the appreciation phase. There are others where the scope for appreciation stabilization
17
is
limited
.
and the policy intervention
is
concentrated in the
first
period of the depreciation phase (ex-post
intervention)
A
3.1
Fiscal Intervention
We consider a government that uses a set of fiscal instruments to afTect the time profile of consumers'
demand. In
particular, the
government can impose a sequence of
linear taxes
{r^
(s*)
,
r^
(s*) }
on
consumers' spending on tradables and non- tradables (a negative tax rate corresponds to a subsidy).
Any
tax revenue
is
returned to the consumers as a lump-sum transfer at date
0, Tq, so
consumers
face the budget constraint
J2P'^{s')i{l
+ r^{s'))c^s') +
{l
+ r^{s'))p{s')c^is*))<Y.P'n{s^)p{s')+To.
t,s'
For a given tax sequence {^t^
the
(21)
t,s'-
economy with no taxes
(s*)
,
t^
(s*) }
we can
define a competitive equilibrium as
in
consumers' budget constraint with (21)
(see Definition 1), replacing the
and adding the condition that the lump-sum transfer Tq
we did
satisfies
the government budget balance
condition
^
We
(3'n [s')
(r^
(.*)
c^
(.*)
+ r^
study a benevolent government that chooses |r^
of the representative
{s')
(s*)
p
(.')
c^
=
(s*))
,r^ (^0} ^o
Tq.
^^ to
maximize the
consumer subject to the constraint of not making entrepreneurs worse
53J]^V(5')c^'M^*)>^,
where
U
is
off:
(22)
the entrepreneurs' expected utility in the competitive equilibrium.
Notice that in this context the choice of linear taxes
the set of feasible policies
and entrepreneurs.
We
is
financial constraint (1).
phase and
later in the
The
not restrictive.
crucial constraint
it
would be simple
The government would
D
in this
economy
on
made between consumers
choose to focus on this exercise for two reasons. First,
in the early stage of the
D
is
the condition that no direct transfers can be
could implement direct transfers,
A
utility
to
if
the government
undo the
effects of the
transfer resources to the entrepreneurs in the
phase, and then transfer resources back to consumers
phase, hence replicating the equilibrium of a frictionless economy.
In practice,
targeted transfers of this kind take place but are limited by a host of informational and institutional
impediments which we do not model
here. In this sense, the set of policies that
the spirit of the financial constraint
(1).'^°
specific
is
more
flexible
respect
than the
forms of macroeconomic interventions that are contemplated in the policy debate. Namely,
^"A "fundamental" view of constraint
in the
Second, the policy described
we consider
form of
financial
payments or
(1) is
in the
that
it is
impossible to extract payments from entrepreneurs, whether
form of taxes.
18
.
most proposed interventions are geared towards increasing domestic savings, thus reducing the
pressure on the real exchange rate by reducing the
we
will see, our
policy-maker
will
demand
and nontradables. As
of both tradables
choose not to distort tradable consumption decisions and
will
only intervene on non-tradable consumption.
In
summary, we consider a general form
of intervention but stay within the boundaries of a
constrained efficiency exercise by ruling out direct transfers between consumers and entrepreneurs.
This choice allows us to identify a basic pecuniary externality, which should play a relevant role
all
practical
3.2
in
pohcy proposals that attempt to curb persistent appreciations.
Policy Perturbation and Pecuniciry Externality
Before characterizing the optimal policy,
us identify the pecuniary externality by stud5dng the
let
impact of small policy interventions around the competitive equilibrium. Consider a planner that
maximizes the consumer's
utility
/
0A {u (c^)
+
u [4))
+
oo
^—^u {cl)
66
+J2l3'u (4,,
J-o
\
where we have normalized expected
taxation problems,
it is
utility
by the factor
easier to characterize the
for tradables
Thus, we
(also multiplying
is
SP
I
in
let
through by
—
As usual
5))}^
optimal
in
terms of equilibrium
quantities,'
the planner choose directly the
(1
—
/3
\
oo
,
+ Pa4 +
(1
and nontradables. Substituting the government budget balance,
the consumers' budget constraint
ca
/?
problem directly
rather than in terms of the underlying tax rates.
consumption paths
—
(1
-,<
+ y. P'PD,A,,
r^cl
P
j=0
(1
—
6))):
oo
<PA + Spy p^PD,j
]
(23)
j=0
J
Relative to the problem of an individual consumer, the planner's problem
is
different in that
it
takes into account the effect of consumers' decisions on p.4 and {pdj}- Consumers' decisions affect
the equilibrium prices by changing the
demand
for
nontradables and, thus, equilibrium wages.
The
entrepreneurs' optimality condition and market clearing in the labor market give an equilibrium
relation between the quantities chosen
by the planner and the prices pA and {pD,j}-
the planner chooses the c^ and c^ , market clearing gives
ua
—
I
—
''See the Appendix for a detailed setup of the planner problem. In the Appendix
4
^^^ "-^J
~
and equal to c^ and c^, respectively,
argument we focus
directly
A
in
the
A
phase and in the
D
features.
19
^
'^Djy
is
constant
phase, and the consumption of non-tradables
phase and only depends on j in the
on allocations with these
^
we show that the second best
allocation shares the following features with the competitive equilibrium; the consumption of tradables
constant and equal to c^ in the
Namely,
D
is
phase. Therefore, also in the perturbation
and entrepreneur's optimality
entrepreneurs cannot be
gives the associated equilibrium prices. Finally, the constraint that
made worse
multiplying through by
off is (also
—
(1
/3
(1
—
(5))):
oo
J/ + 5/3 X; /3^c5^', >
(1
-
/3
(1
-
(24)
f/.
5))
Let us study the effect of stabilizing the appreciation phase, starting from the competitive
equilibrium studied in Section
reducing c^
Specifically, consider the effect of
2.
or, equivalently,
increasing ua, while keeping the naj^s unchanged. This change will affect equilibrium prices and
thus the net present value of the consumer's income. Suppose the planner adjusts c^ so that the
consumer's budget constraint (23)
The
is satisfied.
ua on
following expression captures the marginal effect of a change in
the consumer's
utility:
-6a'u'
— nA) + PA^ +
(1
—
+A
\
where X
—
u' (c^j)
The
equilibrium.
consumer's
first
is
all
first
PDfi
<
row of
order condition in the competitive economy and so
The second row captures
it
It is
equivalent to the
equal to zero at the
is
the net income effect for the consumer.-'^ Since
pA and pofi, and the entrepreneurs'
at date tofi.
consider two cases.
suppose the competitive equilibrium displays pA
First,
< Pa
aiid
(overshooting).
P£,
Let us start with the effect of a unit increase in n,4 on pAto carry
(25)
,
(25) captures the direct effect of the policy.
the noj's constant, this policy only affects the prices
consumption
We
nofi
the Lagrange multiplier on the consumer's budget constraint in the competitive
equilibrium allocation.
we keep
- UA + 135—
On A
duA
an extra unit of ua, then pA must drop
of that unit. Recall that the firm's
If
for the firm to
budget constraint in phase
the planner wants entrepreneurs
be able to finance the extra losses
A
is
{l-{l-6)p)ao=^{pA-l)nA,
from which we obtain:
^
^E±^.
=
oriA
We now turn to the effects of n^
on pd,o- Since pd,o
the entrepreneur budget constraint at date tofl
/
{riDfi
'^Note that the consumer's labor income
net income
is
p
(s')
is
p
(s')
riA)
(26)
riA
< Pq
,
i.e.,
there
is
equilibrium overshooting,
is
=
while
(1
its
n(^s'y
20
-
PDfi) nofi-
expenditure on nontradables
is
p
(s') (1
—n
(s')).
Thus,
—
The
'
entrepreneur's financial constraint
units. In this case, a unit increase in
Wages must
to rebuild to riDfi.
is
ua
rise to
binding and he uses
affects pofi since
compensate
current profits to invest in
all his
new
reduces by / the investment required
it
investment expenditure, so as to
for this fall in
keep the financial constraint exactly binding at npfi. Thus,
dpDfi
f
duA
riDfl
(27)
Finally, notice that in the competitive equilibrium c^^
CjJ^q
=
and the planner
0,
is
unchanged. Therefore, the consumption of the entrepreneurs
sequence 71^^,110,1,
by a marginal change
in
ua and
(24)
is
rise in their
wage
keeping the
unaffected
is
satisfied.
Consumers are hurt by the decline
but gain from the
=
in their
wage
exchange rate) during the
(real
in the first period of the
D
phase.
Which
effect
A
phase,
dominates?
Replacing (26) and (27) in the second row of (25) we have:
- UA + dp—
on a
oua
The inequahty
follows
from Pa
< Pa —
1
=
n/3.0
+
-
1
That
l^^f-
extra term capturing the marginal benefit of increasing
The planner has an
(i.e.
at
is
p/i
+ /3()/ >
in the
is,
ua on
0.
planner'sproblem there
is
an
the expected present value of wages.
incentive to reduce nontradables consumption, so
cis
to reduce the appreciation
reduce pa) and allow firms to keep a larger number of units open, which in turn raises wages
Because
tjofl-
(J)a
>
4'do, reducing wages in
A
generates an excess return in export firms that
transferred back to workers in the form of higher wages at tofi-
and pDfi <
p^jj
consumers are
the expression in (25),
computed
Summing
up,
when pA < Pa
at the competitive equilibrium,
is
positive, so
strictly better off, while entrepreneurs are indifferent to the change.
Consider now a second case, where pA
entrepreneurs are unconstrained at
f/j^o
<
P]^
and pjj^
and hence
Cj^\
>
0.
—
p{^
(no overshooting).
In this case,
This implies both
dUA
and
duA
Replacing these terms in (25) we obtain that the
effect
on the consumer's expected
utility is
negative
-Oau'
given that A
is
that
it
<
(1
- ua) + XpA + Xil-PA) <
0,
XpA- The consumer would benefit from increasing c^ and reducing ua- The reason
makes no sense
for
consumers to cut their wage today
21
if
this action does not raise
wages
which
in the future,
it
when
not
will
there
is
no overshooting to remedy and pp^
is
pinned down by
the technological free entry condition. Instead, the planner, representing the consumers, would like
to exercise its
demand
"monopoly" power during the appreciation phase and
=
given that dcjj\/dnA
is
However
for nontradables.
and violate
f,
no expected overshooting,
multiplier
-04w'
would reduce the consumption of entrepreneurs,
their participation constraint (24).
In fact,
when
there
optimal not to intervene. In this case, there exists a Lagrange
it is
such that the planner's
jj,
this increase
wages by increasing their
raise
(1
first
order condition for
- n^) +
+
Ap.4
A
(1
ua
- p^) +
takes the form
Ai/3/
=
0,
where n^ and pa are at their competitive equilibrium values. The following proposition shows that
no other
feasible intervention
can lead to a Pareto improvement.
Proposition 5 (Constrained
efficiency) If ag
> d^
librium with no taxes solves the planner's problem.
even
if
(i.e.,
optimal not to stabilize the appreciation,
It is
firms are financially constrained and the export sector contracts more than in the
even
Put
for
(no overshooting), then the competitive equi-
if
ao
<
a^).
differently,
if
there
is
no overshooting, there
no intertemporal pecuniary externality
is
consumers, so they cannot trade-off a wage reduction today for a wage increase in the recovery
phase.
The
flip
argument
side of this
consumers underestimate the
It is useful to
is
that
That
is,
the presence of overshooting that makes individual
it is
social cost of their increased
show that the argument
depreciation phase.
just
made
for
demand during
ua can
also be
the appreciation phase.
made
for n/jj
periods t/jj and t^j+i,
Then,
it
is
i.e.,
we
•
further and exacerbate the depreciation
effects of a small intervention are similar to those derived for
marginal
effect of
is still
udj
poj < Pdj+i < Pd
in period j. By doing
•
in
first
order
only affects the current and next period's prices. As before, the
npj
is
Cjj^-,^
—
Cj^-
—
for (26)
0.
Therefore, the
given by
-u'(l-no.M^O.,X^x{^^,r,„^P?^no,»).
Proceeding as we did above
The
ua- In particular, since future n's are
binding after a small intervention, so
an increase
phase
fully recovered in
the consumers accelerate the recovery of the export sector and of real wages.
financial constraint
D
are in the middle of the overshooting phase with
optimal to reduce c^
taken as given, a change in
during the
suppose the planner can only intervene in period j of the
and change n^j by a small amount. Suppose the entrepreneurial sector has not
so,
first best
and
(26),
we
get
^no, + l^^^^nj,,^, = 22
(/
-
(1
-p^,)) +/?/ >
0,
(28)
The
inequality follows from the fact that
consumer's
first
poj <
=
p{j
—
1
order condition, implies that a reduction in
—
(1
/9)
This, together with the
/.
leads to a marginal welfare gain.
p^j
In the competitive equilibrium, the firms' financial constraint depresses labor
non-tradables cheaper and inducing consumers to
demand more
the consumers' reaction to the overshooting and reduces c^
Note that as a
result of this reduction in
what increases
profits
the overshooting
wages. Because (ppj
is
social planner offsets
taxes nontradable consumption).
(it
is
exacerbated, but this
is
precisely
and allows financially constrained firms to accelerate investment. The trade-
between a deeper overshooting and lower wages today
off is
that
c^
The
of them.
demand, making
>
<i>D,j+i^
exchange
in
for
a faster recovery in
reducing wages at to^j generates an excess return in export firms
transferred back to workers in the form of higher wages at to^j+i-
Once we allow the planner
to set policy optimally in
to exacerbate the overshooting in the
D
both phases,
A
and D, some of the incentive
phase goes away, because the planner can already achieve
higher levels of investment by protecting entrepreneurial wealth in the appreciation phase.
interaction between preventive intervention and intervention in the depreciation phase
is
This
a central
aspect of the optimal policy discussion that follows.
Optimal Policy
3.3
We
learned from the perturbation argument above that
the competitive equilibrium
is
if
there
constrained inefficient and there
is
is
an expected overshooting, then-
scope for policy.
We now
turn to
characterizing the economy's dynamics under the optimal policy.
Figure 5 plots the real exchange rate and the number of export units in the competitive equilib-
The
rium and
in the
policy
that the planner attenuates the exchange rate appreciation.
is
effect of
"second best" allocation, in a basehne scenario. ^"^
first
feature of the optimal
This attenuation has the
keeping more export units open in the appreciation phase, and makes the recovery of the
export sector faster during the depreciation phase. In turn, the faster recovery increases the de-
mand
of non-tradables by entrepreneurs
and reduces the extent and duration of the
rate overshooting during the depreciation phase.
final effect of
the intervention
is
exchange
a form of
exchange rate stabilization.
real
Two
rium
robust features of the optimal price path are that pA
level,
and that the increase
decline in pj\
.
The
^^The parameters are the same
the appreciation lasts
Appendix (proof
lower than the competitive equilib-
Pdj's more than
offsets the
price path depicted in Figure 5 also displays a reduction in the initial overshootis less
will discuss other possible cases
when
is
in the expected present value of the
ing at tofl. However, this feature
We
The
real
robust and depends on the economy's
below.
as those used for Figures 2
five periods.
initial conditions.
The optimal paths
of Proposition 6).
23
and
are
3.
As
in those figures,
we
plot the realized path
computed using the characterization
result in the
1.4r
1.2
Optimal Policy
-^
No
Policy
0.8
0.6-
0.4-
_I
I
1
10
Figure
5:
Optimal Policy
24
L_
12
14
Let us explore the optimal paths
phase
(i.e.,
There
Pa < Pa
first
more
in
detail,
by analyzing
the choice of ua) and then the intervention in the
is
phase
< Pd ^^ ^^^
^^'^ Pd,o
A
the naj's).
(i.e.,
an optimal degree of exchange rate stabilization during the
A
Given that
phase.
substitute (26) and (27) in (25), simplify, and obtain the planner's
order condition:
9Au'{l-nA) = \p^A=^i^ +
The
D
the intervention in the
first
as if prices were at first best.
ua
social planner allocates
consumers increase their demand
an appreciation of the
appreciation
is
real
it
In the competitive equilibrium,
nontradables in response to the taste shock, which leads to
for
exchange
smaller than
(29)
l^^f)-
However, due to the firms' financial constraint, the
rate.
would be
in the first best.
This price gap implies that consumers
further increase their consumption of nontradables, at the expense of export units.
taxes consumption of nontradables enough to offset this additional
the real exchange rate and allows firms to maintain a larger
Turning to the
straint
is
D
number
and
in so
doing lowers
of production units open.
phase, notice that along the recovery path the entrepreneurs' financial con-
exactly binding:
=
/ {riDj - noj-i)
until the point
where noj reaches
Entrepreneurs use
all
t
=
present in the second best,
11 in Figure
benefits of the overshooting, which
At the optimum, the argument
is
pn^ <
i.e.,
level, i.e.,
the value
np
that satisfies
- ud) = Xpoand delay
their profits for investment
reached Tld (this happens at
- PDj) noj,
(1
"nondistorted"
its
u' (1
is still
effect,
The planner
consumption
until
they have
Notice also that some amount of overshooting
5).
Recall the argument
pj^.
makes the recovery
subtler since
their
po j
is
faster
equal to
made above on
the social
and increases future values of pojpj-,
.
If
the planner were to increase
PD.o he would have to reduce nofi- But then, since the entrepreneurs are exactly constrained at
tD,i, this
would imply a wage
loss at that date,
i.e.,
pD,i
-
(1
< Pp
For consumers, the net
effect of
a
reduction in nofl would be, rearranging (28),
u' (1
On
the other hand,
would be no gain
in
if
-
nD,o)
-
ApD,o
nD,o)
-
Ap{f
<
A
- / - pDfi +
Pf)
0.
(30)
the planner tried to increase noft, the current wage would drop but there
terms of future wages, given that the entrepreneurs' would be unconstrained
and would employ the extra funds
for
consumption. In this case, there would be a benefit in terms
of relaxing the entrepreneur's participation constraint
-u
(1
-
nofi)
+
XpDfl
+
A
(1
25
and the marginal
effect
- / - pDfl) + nPf <
0,
would be
where
is
^u
conditions just derived
riDfl leads to
show that the planner
an improvement.
p^j —
because
pjj
is
< Vd
^"^
Notice that the reasoning behind these conditions applies only
for all the periods following t/j^o-
some other
where neither decreasing nor increasing
at a 'kink,'
This
optimal policy, the overshooting can only happen in the
PD,j
The two
the Lagrange multipher on the entrepreneur's participation constraint.^''
is
first
key since
it
shows that under the
period of the recovery.
period, then in the previous period
it
If
we had
would be optimal to increase
riDj-i and accelerate the adjustment toward no- Essentially, the optimal path requires that
planner wants to allow for some depreciation in the
D
phase to speed up the recovery,
it
if
the
completely
frontloads this depreciation.
In terms of consumption of non-tradables, a distortion
D
of the
phase (although, not only in the
first
is
also concentrated in the early periods
period). In these periods the following inequality
holds,
u' {1
which can be derived
in
same way
consumption of nontradables
straint
already at
is
PD,j+i
shadow
first
its
best level.
maximum
An
as (30).
\p{^,
individual consumer would like to decrease his
increase npj). However, since the entrepreneurs financial con-
exactly binding, increasing
is
below their
PD.j,
(i.e.,
-UDj) <
noj
any of these periods would reduce the current wages,
in
This has no advantages in terms of future wages, given that
level pjj
.
The
potential cost in terms of current wages (plus the
cost of the entrepreneurs' participation constraint) exactly compensates for the distortion
nontradable consumption.
in
Let us summarize the main results of this section with a proposition. The superscript ce
and quantities
to denote prices
is
used
with no intervention, while starred
in the competitive equilibrium
variables denote the second best.
Proposition 6 (Optimal
Suppose
ficient.
— Pd
P*D
rate in
of the
A
D
(pjf
<
(/
policy) If ao
—
1) / {(3f)
< d^
and
,
nP^
then the competitive equilibrium
< n^
.
Then, the optimal policy has
(relative to the competitive equilibrium), p*^
phase,
P^o < Pd'
<
p^)
"'^
some combination
< p^, some
of both.
the optimal tax on non-tradables in
A
possible to complete this
is
such that
(p'^^
and
argument by deriving the optimal value
of
p*^ (1
n^^.
fi
is
that, given the nonconcavity of the problem, the perturbation
the exchange
lasts at
+ t\) — p^
The
at the
Proposition 6 in the Appendix reaches the same conclusion using a different approacli.
approach
< p^ and
p*^
overshooting in the first period
The overshooting phase
Let us briefly discuss the role of the assumptions on
is
constrained inef-
Depending on parameters, the optimal policy involves some depreciation of
period. If p*dq
^"it
is
first
is
The proof
of
for the different
arguments derived here only yield necessary
conditions for an optimum, while the argument in the Appendix can be used to show sufficiency as well.
26
.
assumption
optimum.
The reason
most one
made
for simplicity, as it ensures that the
second assumption
on consumers
is
is
made
extreme case discussed
to rule out the
>
planner can disregard the constraint pd,j
in 2.3.2,
The
0.^^
where the wealth
effect
so large that the constrained equilibrium displays less destruction than the first
best.i^
For completeness, note that in terms of quantities the optimal policy reduces the fluctuations in
n and the long run
The reason
rate stabilization.
and hence use a
wealth
effect.
The former
export sector.
size of the
the latter result
for
is
and the second best income
if
that consumers are richer in the "second best,"
export sector,
is
uq.
As expected,
for
close to the second best (and they coincide for ao
However,
closer to the first best.
course,
just the counterpart of exchange
To
illustrate this
Figure 6 compares the value of k in the competitive equilibrium and the "second best,"
the competitive equilibrium
is
is
larger share of their labor resources to produce nontradables.
for different levels of financial resources in the
turn
is
result
for
low levels of
ao, the
high values of ag
>
a^), which in
pecuniary externality
is
significant
substantially higher than that of the competitive equilibrium.
the government had effective. direct transfer instruments, then by increasing ao
it
Of
would
be able not only to narrow the wedge between the competitive equilibrium and the second best,
but also that between the latter and the
best.
first
Ex-ante versus Ex-post Intervention
3.4
In our discussion of the optimal policy,
we touched on a pervasive policy concern
any other asset with potential macroeconomic
of an appreciation in the value of the currency or of
consequences
(e.g., real
estate or stocks). Should the intervention take place ex-ante
the appreciation phase) or ex-post
In the example
in the presence
we used
(i.e.,
(i.e.,
during
after the "crash" or depreciation takes place)?
in Figure 5, the
optimal policy involved a combination of both. The
planner stabilized the exchange during the appreciation, but preserved some of the overshooting
in the first period of the depreciation
phase as
both helping export companies accelerate the
well,
recovery.
In contrast, panel (a) of Figure 7 represents a case in which the intervention to offset the
pecuniary externality
is
increasing n^i, increases pd,osatisfies (29),
pDfl reaches
"^The characterization can be
Therefore,
its first
easily
may
"^See the discussion on page 14.
trizations
we have looked at).
same
'''Parameters are the
ao
=
1.1 in
panel
it is
extended to the case
first
attenuation of the appreciation in A, by
possible that, before reaching the level of
best level p^^.
involve a real exchange rate equal to zero for the
to the early periods of D, but
An
done ex-ante. ^'^
entirely
At
this point there
i/i"
>
(/
—
period(s) of the
last
more than one
The
condition holds in
1)
D
is
no gain
for the
ua
consumer
/ (/?/) In that case, the optimal path
phase.
The overshooting
is still
that
may
frontloaded
period.
all
the examples presented (and in
as those used for Figures 5 except for ao
(c).
27
=
all
0.15 in panel (a), ao
reasonable parame-
=
0.5 in panel (b),
0.45
1
1
r,-,,-..
No
Policy
0.44
0.43
-
0.42
0.41
-
/
0.4
/
0.39.
0.38
J
1
0.5
^'
Figure
Consumers' Income and Optimal Policy
6:
from cutting wages further during the
Remember
1.5
A
phase, since this has no effect on wages in the
pA <
that (29) was derived under the assumption that
reaches the level such that
p^^ — pp
D
to the one discussed above in the
Bau'
(1
,
ua
<
s^nd pofi
P£,
phase.
Once ua
the Lagrangian for the planner problem has a kink similar
phase.
A
marginal reduction
in
ua now
gives
- ua)
u' (1
while an increase in
p^,
D
-
UDfl)
-
<
>^Pd
(31)
0,
gives 18
-9au
(1
In this case, the optimal values of
- ha) + XpA + \[l- pa) +
ua and pA
are determined
/i<5/3/
<
0.
by the entrepreneurs' participation
constraint.
The
polar opposite happens in panel
where the policy
appreciation
The
oq,
is
is all
ex-post.
equal to p)j
,
(c)
of Figure 7 (panel (b) simply reproduces Figure 5),
This takes place
for in
if
such case there
the competitive equilibrium price during the
is
no scope
only difference between this scenario and that in panel
which
is
relatively low in panel (a)
See footnote
and high
in (c) (while
14.
28
for intervention
(a), is
it is
at
during this phase.
the level of financial resomxes
an intermediate
level in
panel
(a)
1
1
1
1
1
1.5
-
No
Policy
~\
V
-X
-
0.5i
•
1
2
I
1
1
4
1
10
6
1
12
14
(b)
1.5
/
/
1
^ ^^
/
Z0.5
1
1
2
\
4
-
^
rill
10
6
12
14
(c)
1.5
/
\
V
1
'
^-^
0.5
1
1
2
Figure
7:
1
1
4
6
1
1
10
1
12
Ex-ante and Ex-post Optimal Intervention
29
14
When
(b)).
phase
is
financial resources are relatively abundant, the price-distortion during the appreciation
small and hence the cost of distorting intertemporal consumption by taxing consumption
of nontradables
high.
is
Thus the
social planner opts for
postponing the intervention.^^
Figure 8 generalizes the message of the previous figure and shows the level of p^ and
in the competitive equilibrium (solid)
and optimal policy (dashes)
p^^
wide range of financial
for a
resources ao- At low levels of ao the intervention during the appreciation phase has a large impact
on allocations and there
In fact, the latter
is
is
no need to exacerbate the
overshooting in the depreciation phase.
initial
significantly reduced in this case.
However, as ao
ex-ante intervention and a larger share of the adjustment
Eventually,
when
ao
is
sufficiently large, the policy is
rises there is less
deferred to the depreciation phase.
is
mostly concentrated ex-post, even to the
point of causing an over-overshooting (the region where the dashed Jine
the bottom panel)
equilibrium, there
.
Finally, as ao
is
is
sufficiently high that there
no longer scope
scope for
is
is
below the
no overshooting
solid line in
in the competitive
for policy.
In terms of implementation of the optimal policy in each of these scenarios, Figure 9 reports
the paths of nontradable consumption taxes, r^, corresponding to the three panels in Figure
The pure
7.
ex-ante policy in panel (a) requires strictly positive taxes during the appreciation phase
and a subsidy during the depreciation phase. That
the fact that the pecuniary externality
subsidy component of the policy
is
is
is,
the ex-ante aspect of the policy refers to
entirely resolved during the appreciation phase.
The
simply a mechanism by which, through higher wages consumers
take back any surplus transferred to the entrepreneurs during the intervention in the appreciation
phase. Panel (b) shows the intermediate case, where the exchange rate
the
first
in i/jj
period of the
D
phase. In this case the path for the subsidy
and then converges to
zero.
Panel
the appreciation phase, and instead the tax
(c),
is
is
is
allowed to depreciate in
kept lower in tofl: increases
the pure ex-post policy case has no taxes during
concentrated in the
first
period of the depreciation
phase.
4
Further Considerations for Intervention
In this section
we
briefiy discuss three
important considerations
of the appreciation, distortions in consumers' perceptions,
and
for intervention:
persistence
frictions in the nontradables sector.
''Note also that when export firms have abundant financial resources, there
that any (at least small) intervention can be undone by the private sector (this
30
The
is
is
a sort of Ricardian equivalence, in
an exact result whenever
<l>^
=
0o,o)-
Pa
Figure
8:
Entrepreneurs' Wealth and Optimal Intervention
31
2
4
10
12
14
(b)
0.6
0.4
0.2
-0.2
-0.4
2
4
6
10
12
14
2
4
6
10
12
14
0.6
0.4
0.2
-0.2
-0.4
Figure
9:
Implementation: Optimal Tax on Nontradables
32
Appreciation Persistence
4.1
In our complete markets context, persistence matters only in an ex-ante sense and
the parameter
extreme,
is
On
one extreme,
if
5
is
if
6
low,
is
captured by
close to one (very short lived appreciations) then the
may
be financed are not much and entrepreneurs' internal resources
losses to
units
6.
is
suffice.
On
the other
very close to zero (very persistent appreciations), then the option value of keeping
and there
that policy intervention
is
no reason to protect the export sector
may be
either.
It is for
intermediate
S's
needed.
Figure 10 illustrates this non-monotonicity by showing the region where policy intervention
called for in the (1/5, ao) space.
The shaded
region corresponds to the case where the equilibrium
constrained inefficient and exchange rate intervention
is
general equilibrium effects hidden in this figure.
when
is
is
warranted. Note that there are
For example, as 6 changes, so does
many
Also,
k.
6 rises, firms reluctance to destroy during the appreciation rises. This reluctance exacerbates
the (now shorter lived) appreciation, and hence the resources required to survive each period of
appreciation. However, none of these additional effects
is
shape of the figure and the conclusion that follows from
strong enough to change the qualitative
it.
Medium run
appreciations are most
likely to justify intervention.'^"
Consumers' Overoptimism and Incomplete Insurance
4.2
In reality, consumption binges rarely occur by themselves. In the international context, they often
come
as a response to a rise in national
income due to a positive terms of trade shock
in
commodity
producing countries, or due to a large increase in the supply of capital flows to the country. Adding
external income shocks to our complete markets, rational representative agent setup, would have
no
on the path of consumption, given that consumers
effect
need to add some
One
"friction"
on the consumption side as
extension along these lines
is
fully insure against these shocks.
well.
to replace the taste shocks with
shocks, but assume that either foreigners charge an insurance
-
In an earlier draft
We
premium
income (terms
of trade)
to consumers, or that the
we relaxed the complete markets assumption and studied the polar opposite
case,
where
e.xport
firms only have access to a riskless bond. ,In this context, the export sector resources dwindle as the appreciation
progresses.
The main
policy implication that follows from this modification
D
the timing of the exchange rate
Early on in the appreciation, the optimal policy
stabilization in the appreciation phase.
the intervention to the
is
is
to postpone
much
of
phase. However, as the appreciation continues and the export sector's resources dwindle,
the optimal policy shifts a larger share of the intervention to the appreciation phase (essentially, this amounts to a
gradual leftward movement in Figure 8
.)
33
1.45
1.35
Figure
latter are overoptimistic
10:
Expected Duration and Policy Intervention
with respect to the expected duration of the high income phase A'?^
In either case, consumers find D-insurance too expensive and choose not to insure
is
more complex
and consumers
would be
c^ but
preserved. In particular, nontradables
Of
course,
justified to
also c^.
if
that the basic structure of
demand drops
at the
time of the switch
the social planner does not share in the consumers' optimism, then
implement some
More importantly, even
duration of the appreciation, there
in
is
ignore the pecuniary externality associated to their high expenditure during
still
the appreciation.
it
is
analysis
incomplete insurance introduces wealth and price dynamics
in this case since
within the appreciation phase, however the important point for us
our environment
The
fully.
is
sort of saving policy, with the goal of reducing not only
if
the planner does share consumers' view on the expected
a role for intervention to offset the pecuniary externality, as
our main case.
^'Alternatively,
The extreme
income
we could introduce
procyclical consumption (or short horizons) through non-representative agents.
version of this formulation
in that period.
The
intergenerational transfer
is
one where consumers
mechanism other than through the
the situation just described.
live for
social planner Pareto- weighs a generation
The constrained
real
t
only one period and must consume their
periods from the current one by
exchange rate
goal of the social planner
is
is
available, then
to reallocate
we
/S'.
If
no
are again in
consumption away from non-
tradables during the appreciation phase. Relative to the pure taste shock scenario, a larger share of the adjustment
is
done
in the
A
phase, in order to reduce the burden of the adjustment on the
34
first
generation in the
D
phase.
.
Rigidities in the Nontradables Sector
4.3
Frictions in the nontradable sector generally shift a share of the intervention toward the
For example, this
dollarized.
D, even
in
The
is
typically the case in the
latter
sudden stops hterature, particularly when
A
phase.
liabilities
are
hmits the possibility and desirability of implementing a large overshooting
short lived.
if
Another example
is
the presence of a real wage rigidity, either as the result of a distortion or
of a reservation wage.^^ Yet another
is
that
some
of the inputs of production in the nontradables
sector are tradables.
Let us develop the simplest of these examples and assume that workers have a reservation wage
of
w
units of tradable goods,
Suppose that
below w, but
particular,
this reservation
That
equilibrium.
which
it
is,
is
not binding except, possibly, during the overshooting phase.
binding for the optimal policy but not for the competitive
What
is
like to
bring pj^^
the impact of this binding constraint on the optimal policy? In
of the intervention
is
reallocated to the appreciation phase? Let us return to
the complete markets environment to answer the latter question.
social planner's first order condition in the
9au'
It follows
planner would
in the over-overshooting scenario the social
cannot.
how much
wage
is
(1
immediately that p^''"
<
A
n.4)
phase
=
p^^,
is:
\p^^
where
=
•
A
(1
p^'"'
We know
that in this context the
-.-.
+
66 f)
and
p^**
stand for the second best real
exchange rate during the appreciation with and without a reservation wages w, respectively. The
reason for the inequality
is
that the binding constraint must necessarily lower k relative to the
unconstrained case, and this implies that A
implies that
Ua
increases,
=
u'{6ak) rises with the constraint. In turn, the latter
which given the firms financial constraint can only be achieved with a
larger intervention that drops the real exchange rate
Final
5
Remarks
This paper shows
may
below that of the unconstrained case.
how
financial frictions lend support to the view that persistent appreciations
justify intervention, even
the intervention
is
if
agents are fully rational and forward looking.
not to improve the health of the export sector per
se,
The reason
for
as our social planner
primarily concerned about consumers (workers), but a pecuniary externality within consumers.
is
By
putting excessive cost pressure on financially constrained export firms during the appreciation
phase, consumers reduce these firms' ability to recover once the factors behind the appreciation
^^See Blanchard (2006b) for a
for
more thorough
discussion of
an application to the case of Portugal.
35
wage
rigidities
and appreciations; and Blanchard (2006a)
The
subside.
a severe overshooting and real wage collapse at that stage, which hurts
'result is
consumers more than they gain from the extra consumption during the appreciation.
the perennial policy problem of the timing of inter-
Our normative framework sheds hght on
vention.
We show
that, other things equal,
during the appreciation phase, then
it
is
if
optimal to intervene ex-ante.
sector has substantial financial resources (although not
ex-ante intervention
is
on the export sector are
financial constraints
either costly or ineffective,
and
enough to
it is
Conversely,
if
tight
the export
then
fully finance the recovery),
optimal to postpone intervention until
after the "crash."
In abstract, the optimal policy can be implemented through an appropriate sequence of taxes
and subsidies on nontradable consumption.
leaving to expenditure policy
In reahty, the flexibihty of such policies
and central bank's reserves management most
these are not perfect substitutes for taxes and subsidies,
much
is
limited,
of the burden.
of our insights
still
While
carry over to
them.
Let us conclude with a few clarifying remarks and extensions.
is
When
thinking about policy,
worth noting the distinction between an "appreciation" and an "overvaluation." The
an elusive concept
to a situation
in practice
but
it
is
not limited to an appreciation episode as
rigidity
is
is
higher than
it
it
is
However, this gap
socially optimal.
an example of an overvaluation during the depreciation phase.
The
latter
example
is
can also take place during a depreciation phase. The
an example of when such overvaluation cannot be cured
the depreciation phase.
is
has a well defined meaning in ours: an overvaluation refers
where the exchange rate
over-overshooting result
latter
it
fully
A
wage
with intervention within
also illustrates the role of early intervention in limiting
future overvaluations.
We
note that our model uses a single reason
in the appreciation
and depreciation phases.
scenarios as well. In particular,
for a technological
we could
— a financial friction—
constrained production
However, some of our conclusions extend to other
replace the financial constraint in the depreciation phase
time to build assumption. In such case, the overshooting
to excessive export destruction in the appreciation phase
The main
for
difference in this instance
is
and there
is
is
also directly linked
a reason for intervention.
that the optimal policy does not prolong the intervention
into the depreciation phase.
Finally, while our anatysis focuses
on the
portant relative prices within an economy.
cost consequences for sectors that
production.
More
broadly, ours
is
real
exchange
rate,
it
seems suitable
For example, a real estate
boom
compete with the construction sector
a model of the optimal
the presence of temporary (but persistent) shocks,
technological flexibility.
36
management
when some
for other im-
can have important
for inputs
and
factors of
of sectoral reallocation in
sectors have limited financial
and
Appendix
6
Proof of Proposition
6.1
The
cutoff
is
1
given by
a^'
where p^ ,pp
and n{, are defined
,
in the text.
Let us conjecture and verify that
and quantities form an equilibrium. Given the conjectured prices
prices
and
,n-'^
1-/3(1-5)
verifying) that
V {a,n~;s*) =
a
+
q{s')n~
(i.e.,
i/;
=
(s*)
and
entrepreneur's optimality conditions (9)-(ll) shows that the entrepreneur
all
feasible choices of
consume the
n{^)
for
-
(1
-PdWd
each history
s*
~
[A,
left
a^'',
=
for each history s*
main
inequalities
text,
is
that
1-/3(1-5) <
{A,A,...,A}; set n(s*)
n^ >
=
1,
- P)Oa +
<
1,
and x >
0.
5/3)
he can
ao,
=
/(n^ —
n{} a ({s\ D))
,
and 1/
(1
n-^^ in
+
=
One
clearing.
This follows from substituting
0.
6 a/ ((1
among
at each s', indifferent
D}. These decisions are consistent with labor market
aside in the
and using the
definition of n{^
show (by guessing
,
=
...,
a^^ these
Then, inspecting the
1).
,
D,
...A,
is,
=
>
ao
and {a ((s*,St+i))}j
^^ If the entrepreneur begins with
and then adopt the following rule: set n{s*) = n;^ a((s*,£>))
,n{s*)
a-''''
and a{{s\A))
=
which we
final check,
c^'*^ (s')
difference ao
possible to
it is
(p{s*)
if
the
SPf)
<
H (n)
>
1.
Proof of Proposition 2
6.2
First,
we
Lemma
establish a preliminary
lemma.
Define the function
1
H{n) = fn- (l
1
the equation
for each
H (n)
n > n*
one of which
.
=
has a unique solution n* €
The solution n*
is 0.
is
(0, 1),
-n,
Jor each k
increasing in x. If x
=
>
Moreover,
the equation can have one or two solutions,
In this case, the properties above apply to the largest solution.
[0, 1), H (0) = —x and lim„_^i H (n) = oo. Consider
—
the case x > 0. Let n* be a solution, then /-(I
k/(1— n*))>0 must hold. If n > n* H' (n) =
follows from / - (1 - k/ (1 - n)) > / - (1 - k/ (1 - n*)) > 0. This
/ - (1 - k/ (1 - n)) + Ku/ (1 - n)^ >
A
Proof.
solution exists because
H
is
continuous in
,
implies that
H {n)
>
for
each n > n*, and the solution
respect to x follows from the implicit function theorem.
is
another solution n*
The proof
will
some properties
>
0,
proceed in three steps. First, we define a
of this
1.
unique.
x
=
The comparative
the solution n*
=
statics result with
is trivial.
the properties stated can be proved following the steps of the case
map.
Finally,
we show that
we can construct an equilibrium with the desired
Step
is
When
Fix a value for k G
[O,
k^^]
this
map
map T
for the coefficient k.
has a unique fixed point.
properties.
and construct an equilibrium
37
as follows.
.r
>
If
there
0.
Second, we derive
From
this fixed point
Phase A.
li
(l-/?(l-<5))ao>(p^''-l)h-^j,
then set pa equal to p^
,
n^ =
set
follows
>
0.
<
Since k
— kBaIVa
^'^'^
= ^[(l-^(l-<5))ao-(p^'-l)n^] >0.
aD,o
Notice that n,4
1
(32)
k^^
we have
1
— kBa/pa >
1
- k^^9a/Pa >
(33)
Oj
where the
last inequality
from assumption (Al).
If (32)
does not hold, then set pA equal to the solution of
{pA-l)(l-^Y
{l-P{l-6))aQ =
(which has a unique solution in
the right-hand side of (34)
is
[I,
Pa
]))
ua =
set
pa £
zero, therefore
— k9a/pa and
I
=
ao,o
and ua >
k9^,p;^'
(34)
0.
Notice that when pa
=
k9a,
0.
Phase D. Define
Pd
Construct the sequence {noj} that
satisfies:
=
JinDfi-nA)
1
~
:;
i
=
f {riDj -riDj-i)
until
nD,j+i
is
larger than
1
no. From then on
-
=
aofl
+
/tia,
solution with the largest
Lemma
n^
o).
1
1, 2, ...,
>
for all j
J
(36)
J.
ensures that (35) has a solution for riD.o
To show
riA
that nofi
(if o,d,o
+ I^a =
consider the following: Either
>: ?^,4
economy converges
H {no)
to n£) immediately
0,
<
0,
pick the
and the
and
1
dA
<1- li-TT
<l- K-j^ =nD,
Pd
Pa
in the
=
(35)
set
solution will be larger than n£). In this case the
where the inequality
od.o
no.j for j
:;
noj = no
Letting x
+
- nofi JWd,o
middle follows from assumption (A3).
If,
instead
H (np)
>
H (7^D,o)
=
0.
In a similar way,
it
then
'
Notice that
H (ua)
where the inequality
in the
=
i
—
,
IjriA-
aD,o
l) '^A
-
aofi
<
middle follows from
K
l-UA
k9a
1
possible to prove that (36) implies n^^j
^
^
1
/&
<Pd <^
I-uaOa <PAir
TA
(the second inequality follows from (A3)). Therefore,
is
< [Pd -
> npj-i
/-b
Lemma
for
38
each
1
j.
1
1.
implies that no^o
> n^.
R-ora these two steps
to obtain n'
Step
map T
This defines a
.
It
2.
we obtain a sequence
[O,
:
map T
can be shown that the
/v'
In the construction in Step
+ fn^
Ua)- But since
n^ =
is
1
notice that
independent of
— 9ak/pa, noj =
than proportionally than
less
Step
ti.
Define the following
3.
map
Step 2 shows that this
point (uniqueness
is
+ A))<
if
1
{1
+ A)T{k).
k leads to a (weak) reduction
(32)
if
in
ua and noj
satisfied, then, using the definition of Py^
is
(32)
—
in expression (3),
continuous. Furtliermore, let us prove that
is
in
k;
{pdj}, which can then be substituted
[0,k^''].
r(/v(l
an increase
1,
D
conditions of phase
initial
show that aoo
=
p/i,
—
«'''']
for all j (for the
,
it is
k,/pd,j, this implies that the prices
pA and po^j must
increase
Therefore, k! increases less than proportionally.
map
for z
=
log {k)
:
continuous and has slope smaller than
1.
Therefore this
map
has a unique fixed
not needed for the statement of this proposition, but will be useful for the following
is
Let k be the fixed point and consider the prices and quantities constructed in Step
results).
possible to
not satisfied, then an increase in k leads to a decrease in
is
that they are an equilibrium,
it
To ensure
1.
remains to check that the sequence of prices and quantities are optimal
money
the entrepreneur. Derive the marginal utility of
—
J_
;r~T'i*'Dj+i''Dj^^Trm
7-(l-PD,
By
construction
we have poj < Pd
and cash savings, apj+i, are zero
which implies that (ppj >
,
until the point
for
at tpfi from the recursion:
where (ppj
1.
(37)
Moreover, entrepreneurs' consumption
=
1.
iff
pa < Pa
To check optimality
in
phase
.4,
notice
that
4>A-^^^,4>O,0
and
(f)^
either
>
pa
(pQQ
or
ififp/i
< Pa
some pd,j
will
,
and, by construction a^fi
be
strictly
below their
zero
is
Notice that as long as ao
best value. Therefore, i^^
>
<
a^^
1.
Proof of Propositions 3 and 4
6.3
The
following
Lemma
2
lemma
provides a useful preliminary result.
The equilibrium value of k,
is
non- decreasing in uq.
Proof. Let T(K]ao) be the mapping
by the
of K.
aj),
first
initial
Now,
wealth ag. Choose two values Oq
fixing k'
then an increase
If (32)
[O,^-'^'']
we want
to
in oq leaves
show that
T
<
is
—
a'^.
k'
it
defined in the proof of Proposition
indexed
T(«;';ao). If (32) holds at
increases aD,o (from (33))
and leaves ua unchanged.
in oq,
i.e.,
a£)fi leads to an increase in pA,
39
2,
and n" be the corresponding equihbrium values
>
monotone
pa unchanged, and
does not hold, an increase in
[O, /x^''l
Let
r(«;';aQ)
and an increase
in a^fi
+
/ua, since
.
either remains zero or
ao
means
that, in phase
T
it
will
>
= p'^
=
k'
in equilibrium.
follows that at the
in
new
Consider next Proposition
4.
for all j, and, thus,
friA increases. This
a (weak) increase
>
k" by construction, this implies k"
,
Consider
This means that (32) holds at
pA = Pa ^^^
equilibrium
noj
+
This impUcs that r(K;aQ) has a fixed point in
.
T (k"; Qq) =
can prove the two propositions.
In both cases, aD,o
increases.
be a (weak) increase
T(K';ao)
has a unique fixed point and
Now we
p^
D, there
Therefore, T{K';aQ)
for all j.
Since
becomes positive and ua
Proposition
first
Cg. Since
>
o.d,o
>
n"
k'
.
Suppose that
3.
n', (32)
in
poj
[k',k^''].
U
at a^
we have
holds a fortiori for o'^q,k",
0.
po^ = p^
Suppose that at Uq we have
in equilibrium.
This means that
the following inequality holds
where
n'^
=
-
1
k'/p{,
.
Now we want
- r^-T-)
(l
^
^^'d
holds at
a'^
p'0
•
=
M
does not hold at ag, then we have
Notice that k"
<Pq-
>
+
>
«D.o
Ha +
n'
+
(38)
flD.o
+
a'^ g
a'D.o
=
>
=
;^
(39)
o-'ofi-
then some algebra (using the definition of p^
/< + <,o
If (32)
M
to prove the following inequality
fn'k
If (32)
+
ri'o
[(1
)
-
o-'do-
shows that
/3
(1
-
ao]
(5))
>
n'j^.
p^, at ag,
and
Furthermore, we can show that
holds because, on average, equilibrium prices are larger. However,
p^ =
n'4
This implies that
Pa ^ ^
~
p'a
and since n^
=
1
also holds at Og,
— k/pa
this implies n'^
and therefore po^g
proof of Proposition 3 and show that
6.4
The
= p^
a^
g
>
'
1^'
n\. Therefore, (39) holds in
Notice that
.
> a^
q
and
if
n'^
all
cases.
(32) holds at ag then,
=
This implies that (38)
we can proceed
as in the
tt-'a-
Setup of the Optimal Policy Problem
set of feasible allocations
Definition 2
(Feasibility)
quence of tax rates [r'^
and quantities {p
rium under
We now
(s*)
is
defined as follows.
The allocation {c^
(s') ,t'^ («')},
(s')
,c^
(s')
,
c-'"''^
,n{s*)}
(s')
is
feasible iff there exists a se-
wealth levels {a{s')}, and prices {pis*) ,g(s*)} such that the prices
,q{s*)} and {c"^ (s')
,
c^
(s*)
,
c^''^ (s*)
,n{s*) ,a (s*)} constitute a competitive equilib-
the tax rates {r-^ (s*) ,t'^ (-s')}-
derive three necessary conditions, (40) to (42) below, that any feasible allocation
must
satisfy.
Then, we define the problem of a planner that chooses an allocation subject only to the consumer's budget
constraint, the entrepreneur's budget constraint (7),
and conditions
the original planning problem, given that this set of constraints
40
is
(40)-(42).
This
is
a relaxed version of
necessary but, in general, not sufficient
for feasibility.
We
6
we make use
perform a cliange of variables that makes the
also
problem and we derive
first-order conditions
which are
sufficient for
rela-xed
planning problem a concave
an optimum. In the proof of Proposition
of these first-order conditions to find allocations that solve the relaxed planning problem.
First, notice that the entrepreneur's optimality implies that a feasible allocation
must
satisfy the condition
(g(.')-(l-p(5')))n(s')>/3 J2 ^ist+i\st)q{{s\st+i))n{s').
To prove
this inequality, multiply
by n{s^) both sides of
(40)
and use the complementarity condition to
(9),
obtain
(g(s*)-(l-p
(.'))) </,(s*)n(s*)=/3
^{st+i\st)<P{{s\st.+ i))q{{s\st+,))n{s').
Yl
St
+ lGS
Moreover, the entrepreneur's optimality condition
for a(s*) implies that
(p
{{s*
,
>
St+i))
4>{s*) for all St+i-
Substituting in the equation above, gives (40).
Second, recall that q
{{s''
,
<
St+i))
f which implies
'
q{{s\st+,))n{s*)<fn{s').
(41)
Finally, define the function
= /max{x,0},
G(a:)
for a generic variable x,
and notice that equilibrium
in
the adjustment sector implies that, for
all s*
and
St+l,
q {{s\ St+^)) {n {{s\ St+,))
We
perform a change
-n
= G
{s*))
-n
{n ((s*, 5t+i))
{s'))
(42)
.
in \-ariablcs, defining
z{s')
^
{p[s^)-l)n[s^)+q[s')n{s^),
j/(.')
^
q{s')n{s'-'),
and
for all consecutive histories s*~^
s*.
Substituting the government budget balance in the consumer's budget constraint (21), and using the
market clearing condition C^
(s*)
^
=
1
/3*7r
-n
{s*)
(s*),
c^
we obtain the budget constraint
(s')
<
^
/?V
(s*)
{s')
n
[s')
(.')
-
G {n
V
.
Substituting z and y and using (42) on the right-hand side we get
^/3'^
{s*)
J
(s')
= ^/3V
-
(.') [. (s')
y
(.')
+n
(.')
-n
(s'-^))]
.
(43)
Substituting z and y in the entrepreneur's flow of funds constraint gives
c^-' (s')
+
z [s')
-
y
(s')
+0 Yl
41
^i't+i\st}a {{s\ St+u))
<
a
{s')
.
(44)
,
The
to
is
relaxed planner problem
to choose sequences {c^ (s') ,(F'^ (s*) ,n(s*) ,a{s*))
is
maximize the consumer's expected
a concave problem, so the
the solution
is
first
terms of variables indexed by
constraint,
maximizes
A
stationary in phase
To
of periods since the transition.
A
and
utility subject to (22), (40), (41), (43),
order conditions are sufficient for an optimum.
and
that, in
(44).
It is
and {z{s^) ,y{s^)},
The
relaxed problem
possible to
show that
phase D, quantities and prices only depend on the number
we write the problem
save space and help the interpretation,
and {D,j). Moreover, we normalize the
and the entrepreneur's participation constraint by the constant (1-/3(1
—
directly in
consumer, the budget
utility of the
6)).
Then, the planner
-
oo
dA
[u (1
-nA) + u
(c^)]
+ 5PJ2i3'Sd
[u (1
-
nD,j)
+ u{clj)]
,
j=0
subject to
oo
^A
oo
+ Yl l^^'^D,j +G{nA-
n_i) + S0G (n^.o -
tia)
+
<5/3
j=o
^Z /^^^ i'^Dj - noj-i)
j=i
oo
<
{zA
- y.A + n^) +
- VDj +
5/3^/3-' {zD,j
riDj)
(A)
oo
Ca'^
(1-/3(1-
5)) ao
Next
<5/3
l35aD,o
E
+
/^''^dj
+
yD,0
odj - 0aDj+i +
yD,j
-
(40)-(41),
to each constraint
Pao,!
>{l-P{l-S))U,
Va - za -
-
0-D,Q
and conditions
—
-
+
c^^"
ZD,0
—
C£)'
ZD.j
-
Cjjj
>
^
>
(m)
0,
(^a)
(45)
0'
(SPVD.O)
(46)
{6P'+'vD,j)
(47)
for j
>
1,
which take the form
Va
<
fuA,
VD,j
<
fnD,j
ZA
>
SpyD,o,
ZD.j
>
Pyoj+i
(7^)
for all j,
iSP'^'lD,)
iVDfl)
for all j.
we write the respective Lagrange
iSP'^'VDJ + l)
multiplier.
Let us take first-order conditions with respect to n
-^^u'(l-n.4)-WA +
A
+
with respect to y and z
=
0,
PSp^r^a^^
=
0,
X-UA+Vofi
=
0,
X5P^+'-5/3' + 'i^D,j+5l3^+'vDj+i
=
0,
-X + ua-'Ja
-X5p^+'
+
SP^+'uD,j
- 5/3^+Sd,, -
42
/7.4
=
0,
and with respect to a and
c-^'^
^Dfl
>
{aofi
>
0),
(48)
VD,j+i
>
(acj >
0),
(49)
II
>
^A
(c^''>0),
M
>
^D.j
+
-i^A
+
-i^Dj
Rearranging these conditions shows that a
multipliers X
<va ^
i^Dfi
<
'^D.i
< • <
an optimum
is
that there exist Lagrange
W) + /(i>.4-A)
=
0,
(52)
+ A(l-(l-/3)/)+/(i.j3,o-z^.4)
=
0,
(53)
-eDu'{l-nD,j) + X{l-{l-p)f) + fii^D,j-'yDj-i)
=
0,
and conditions (48)-(51) are
+
A(l
+
and 6
5
prove Proposition 6 by giving a complete characterization of the optimal allocation.
In the
first step,
we
define two
construct a candidate optimal allocation and
a side product of this step.
is
exchange rate and
Step
that
1
+
1.
We
for the
<
1
+
maps J and
we show
In the third step,
we
J.
is
is
split
we use these maps
In the second step,
that this allocation
The proof
indeed optimal.
The
to
proof of
derive the implications for the optimal path of the
optimal tax in A.
define the two
5l3f/4>''X
(54)
.
satisfied.
Proof of Propositions
in three steps.
5
(51)
such that
M.
-eoti'(l-nD,o)
We
(cS5>0).
sufficient condition for
-6i.4w'(l-n.4)
6.5
(50)
5/3/
maps J and
=
p^^
J:
since 4>a
Define the
>
map J
:
For any pA e
1).
[1
[I
+ 5jif /(jf^,pj^
+ 5 P f / (P"^ p^^]
]
,
^ M as follows
(notice
find the unique ^ that
solves
=
^
where pD,o
=
1
—
/
+
^l^p]e'f+5peo r-'""^
^P"
p/
{4>'a
{va
—
1))
n>i
,
+
^'^
i^D,onn,o
+
E
P'p'Dnn,:j
and the sequence {ua, {noj}]
(55)
,
j
is
given by
= -^(l-(l-(5)/3)ao,
PA-i
(56)
riDfi
=
-^0$f(l-(l-5)/3)ao,
riDj
=
mm{P~^ no fi, no]
for j
>
(57)
(58)
1,
where
no =
To show that such a
£,
increasing function of
^,
exists
and
is
1
- C^D /?{,'•
(59)
unique, notice that the right-hand side of (55)
and ranges between a positive value,
43
at ^
=
0,
and — oo
for ^
is
—
»
a continuous nonoo. Set
J
(pa)
=
^
J
(the function
J
{pa)
>
is
allowed to take negative values but we will see below that at the relevant values of pa,
Combining the terms containing pa on the right-hand
0).
P-4
which
is
_
(1
-I
PA
(1
_ 5)^)^0 +
map J
Define the
[0,(1-
:
-
(1
-I
PA
J ^P~f
4>A
Applying the implicit function theorem,
in pA-
ao/(p^^-l)] as follows: For any
5) ;3)
we obtain the expression
fl-/+_l_^)-i_^-(l-(l-<5)/3)ao,
5/3
V
monotone decreasing
side of (55)
n^ G
it
<
follows that J' [pa)
-
[0,{1
-
{1
S)
0.
p)ao/{p{^ -I)]
find the unique positive f that solves
1-/3
+ SP
p{^nA
=
where po,Q
such a
(^
Step
1
—/+
and
exists
is
and the sequence {tidj}
,
/3//(?!>5f
PDfiTiDfi
J
unique. Set
=
(j^a)
kJ{p/
•"
we know that L(pa)
1,
-
+ 5Pf /cj}"^) >
(1
0,
L
or
in detail the first case,
(p;^
j
<
^
an increasing function of p4. Therefore, three mutually exclusive cases
is
Either there exists a unique
are possible.
L
(n^i)
easy to show that
0.
^(l-(l-5)/?)ao.
;r -IP^
P^A
step
it is
>
Define the function
2.
L{pa)^1R"om
given by (57)-(58). Again,
is
immediate to show that J'
It is
C-
+ ^P^Ponoj
We
0.
pa £
+
[1
^PII^'a^Va
\
that solves the equation
can construct an optimum
which correspond to the case depicted
in
for
Figure
We
each of these cases.
Let p\ be such that
5.
=
L{pa)
0, or
will analyze
L
{p\)
=
0.
Set
_
,
the assumption
(/
—
1) / (/S/)
4>'^^
<
(/
—
the argument needs to be
requires a slightly
more involved
= J {p*a)
and
set
c^
=
61,4^*
amended
5{(iff
(Pa
Pa-^
>
q
0,
given that
to allow for a
number
p^ <
= p^
= OdC-
p^
Set
and c^
Finally, the values for the entrepreneur's
for all j
>
1,
g^
=
and
0,
argument
q*p j
consumption are
(l-;3(l-<5))ao
z'^,
we need
A*
=
{zp
},
to find
Cd,0*
=
(1
-PD, 0)^^0,0 - /("D,0 -".4),
Cd,T
=
(1
- PD,j)nD,j - I ("Dj - nuj^i)
we can
and show that we have found an optimum
Lagrange multipliers A*,i/^, {i^p A, and
1/^*. Notice that the condition
L
{p*A
=
A*p^
>
0, this
analogous to
for all j
>
0.
Let
+ (l-p.4)n^,
for j
>
I.
Given the construction of the sequence {n\, {n'j^A}, entrepreneur's consumption
and
f
is
(ji^f
set as
=
allocation,
=
po j =
Set the sequence {'^^ij'^Dj}} according to (56)-(58).
c^"
Having defined a candidate optimal
In the case
p^^.
of periods in which
definition of the function J, but otherwise the
the one for the case analyzed here.
1^*
ensures that p*^
1) / (/?/)
1
is
always non-negative.
define the corresponding sequences for 2/^,
for the relsLxed
fi*
problem defined
in 6.4.
such that conditions (48)-(54) are
{vdj}
To do so,
satisfied. Set
can be re-arranged to give
= 9V(l-n^).
44
(60)
.
Moreover, by construction
n*Dj
than some J*. This implies that
for j greater or equal
with equality
< l-r0D/p{fforallj,
X*pi,'<eAu'{l-n},^^)
with equality
>
for j
J*
.
Then we can
=
I'hfi
By
construction
satisfied for all j
t^A
be constant
will
z/Jj
set i/^
+
j
A and
[Odu
-
(1
nj, o)
> J* and we can
for j
and we can check that
=
Cp" >
as
an
equality.
Some
The consumer's budget
v*^ j.-
when
(/|p
This confirms that (51)
=
and {pd
L and J
,}, the
entrepreneur's expected utility
U
fi*
consumer's budget constraint and
constraint can be rewritten as
equation (55)) guarantees that this condition holds
(in particular
+
<5/3
2^ /?^Co,,
-0
=
/
—
3Y
^^-^^o/-(l-Pb.
11
right-hand side of this equation
is
-,
equal to
-ao.
cp'^^ao,
which
is
equal to the
in the competitive equilibrium, since
C/
=
7^
4^n
P$f-l,Vo/-(l-,/-^)
r«o
=
(61)
^$rao.
This completes the argument that the candidate allocation solves the relaxed planning problem.
to
show that
is
lengthy but straightforward algebra, using the flow of funds constraints, shows that
C.4
p*^
i.e.,
>
oo
^-^(l-'^H
Given the prices
,
=
satisfies the
oo
the construction of the functions
- ^*Pd)
set fi*
> J*
only for j
Furthermore, we can check that the proposed allocation
the entrepreneur's participation constraint.
for all i,
this allocation
is
feasible.
To do
so,
..
we
first
derive values for the
0^ and
4>*£,
,•.
remains
It
We set 0^ =
cf)'^,
Pf
TT
j=of- (1-Pd,
and
(10)
(p*Qj
and
=
j
and
for all j
>
1.
(11) are satisfied.
while c^'^*
all
1
=
t'^*
c^'q*
=
These, can be used to check that entrepreneur's behavior
To check these conditions
and a^
.
=
for all j.
notice that
0^ >
(^^ q
> 0o
i
is
and
Finally, the tax rates are set as follows:
optimal,
4>*d,j
rj*
=
—
i.e.
that
^ fo'^ ^1^ J>
r^*^-
=
for
and {t^'A are such that
eAu'{l-n*^)
=
A>^(1+t:i),
eD«'(l-n^J
=
A*p^,,
45
(1
+ rBj.
(62)
(63)
Let us discuss briefly the cases wlicre
now
that condition (60)
L
+
(1
and L (pa') <
>
SPf/^)"^)
we have
holds as an inequality, and
>
i^^
^^ the
0-
The
A*.
first case,
we have
rest of the construction
is
analogous to the one derived above. In the second case, we make use of the function J to find the value of
1^*.
In particular, define the function
L{nA) =
l
riA,
71
Pa
and find an
L
(p^^)
£ [0,(1-
n*i
and L
<
but we have
n^ <
(1
-
(0)
>
0).
—
(1
—
(1
(5)/3)
Then,
5)
L
ao/(p^^-l)] such that
C
set
P) ao/(p^
= J (it-a)
—
=
(n^)
and A*
=
(notice that
L
(^(1
-
-
(1
5)/3) oo/(p^''
1/^*. In this case, (60) holds as
an
so the entrepreneurs have positive financial savings
1),
- r
equality,
when they
enter phase D,
al,^o
This
is
with
a*jj Q
consistent with feasibility, given that
Step
>
3.
allocations.
it
= {'^-i^-S)P)ao-{p{'-l)n*A.
0).
The
That
We
rest of the
<
p*^
want to show that
follows from (61)
and
p'g^^
<
^
^o ^hat
=
(J>*a
cp}) o
(which implies that (11)
is
satisfied
proof proceeds as in the baseline case.
<
and p^
p;^
= Pa
pa
p*^
<
follows immediately from the construction of the optimal
Pj~,
Consider
p'^/.
-
p{^ that (p^=
first
the case where
=
1) / (5/3/) (^"^
Y[ Pf/
(/
p^
-
=
(l
1
+
SPf/cp'^A- ^^ this case,
- Pd,j)) >
1.
This implies
j=o
that pj^
>
p^.
^(p^) <
0,
which implies
Next, consider the case where
ni^
=
P/i
first
>
p_4.
+
SPf/cp'^.
Pa
Pa
In this case,
(recall the construction of the
we have, by construction
it is
possible to
,
equilibrium in Proposition
2).
(64)
^
show that J (p^) < k^''.
Therefore, consider the case p"^ < p^ where n'^ =
inequality follows because
ately follows that p'^
1
l-^^<l-^^^<^(l-(l-5);3)ao,
~
Pa
where the
>
If p'4
(1
—
= p\
(1
-
n'^
< n^ and
< Pd
implies p*j
The assumption
then
5) /3)
it
ao/
immedi(p,4
—
1)
inequality (64)
imply
{l-il-S)p)ao ^ {l-{l-5)P)ao
PA -I
Pa - 1
giving the desired inequality.
Let us derive the optimal tax r^
Also by construction, whenever p\
n\, which, together with
whenp^Q <Pd- By
>
l
+ 5Pf /(jfA
(62), implies that
p\
(1
construction,
p^
q
the following condition holds as an equality
+
t\)
46
=
Pj^
.
> l + SPf/cp'^.
— OaC/Pa ~
'^
References
Aghion,
[1]
P.,
Bacchetta, and A. Banerjee, "Financial Development and the Instability of
P.
Open Economies," Journal
Aghion,
[2]
Monetary Economics, 51
ductivity Growth:
The Role
of Financial Development,"
Blanchard, O.J. "Adjustment within the Euro: The
[3]
(6),
pp. 1077-1106,
September 2004.
Bacchetta, R. Ranciere, and K. Rogoff, "Exchange Rate Volatility and Pro-
P.
P.,
of
mimeo. Harvard, 2006.
difficult case of
Portugal,"
MIT
mimeo,
February 2006a.
[4]
Blanchard, O.J. "Current Account Deficits in Rich Countries,"
[5]
Burnstein, A.,
,
[6]
M. Eichenbaum, and
S.
MIT mimeo,
October 2006b.
Rebelo, "Large Devaluations and the Real Exchange
Rate," Journal of Political Economy, 113(4), pp. 742-784, August 2005.
Caballero, R.J. and A. Krishnamurthy, "International and Domestic Collateral Constraints
in
a Model of Emerging Market Crises," Journal of Monetary Economics 8
(3),
pp. 513-548,
2001.
[7]
Caballero, R.J. and A. Krishnamurthy, "Smoothing
ory 119(1),
November 2004,
Sudden Stops," Journal of Economic The-
pp. 104-127
[8]
Chaney, T., "Liquidity Constrained Exporters," mimeo. University of Chicago, 2005.
[9]
Corden, W.M., "Booming Sector and Dutch Disease Economics; Survey and Consolidation,"
Oxford Economic Papers 36 (1984), 359-380.
[10]
Fitzgerald D. and K. Manova,
"Why
Don't Exports Respond to Large Devaluations?" mimeo,
Stanford, 2007.
[11]
[12]
Farhi, E.,
M. Golosov, and A. Tsyvinski, "A Theory
and Regulation
of Financial
Intermediation,"
MIT mimeo
Geanakoplos,
and H.M. Polemarchakis. "Existence, Regularity, and Constrained Subopti-
mahty
J.
2006.
of Competitive Allocations
rium Theory.
[13]
of Liquidity
Golfajn,
I.
Vol.
2.
ed.
When
the Asset Market
by Gerard Debreu, 67-97, 1996.
Is
Incomplete." in General Equilib-
(First published 1986.)
and R. Valdes, "The Aftermath of Appreciations," Quarterly Journal of Economics
114(1), February 1999, pp. 229-262.
[14]
Gourinchas, P.O., R. Valdes, and O. Landerretche, "Lending Booms: Latin America and the
World," Economia, 1(2), Spring 2001.
47
[15]
[16]
Panizza and R. Rigobon, "The Long
Hausmann,
R., U.
Volatility",
Jounal of International
Krugman,
P.,
Money and
Run
Finance, 25
Puzzle of the Real Exchange Rate
(1),
(2006), pp. 93-124.
"The Narrow Moving Band, the Dutch Disease, and the Competitive Conse-
quences of Mrs. Thatcher," Journal of Development Economics 27 (1987), 41-55.
C,
"Inefficient Credit
Booms,"
MIT mimeo
[17]
Lorenzoni,
[18]
Manova, K., "Credit Constraints, Heterogeneous Firms and International
2006.
Ti^ade,"
mimeo,
Harvard, 2006.
[19]
Wijnbergen,
S.
van, "The
Dutch
Disease:
A Disease After All?"
1984.
r
3521
48
The Economic Journal, March
Date Due
Lib-26-67