WPS4926
P olicy R eseaRch W oRking P aPeR 4926
Optimal Devaluations
Constantino Hevia
Juan Pablo Nicolini
The World Bank
Development Research Group
Macroeconomics and Growth Team
May 2009
Policy ReseaRch WoRking PaPeR 4926
Abstract
According to the conventional wisdom, when an prices to characterize optimal fiscal and monetary policy
economy enters a recession and nominal prices adjust in response to productivity and terms of trade shocks.
slowly, the monetary authority should devalue the Contrary to the conventional wisdom, in this framework
domestic currency to make the recession less severe. The optimal exchange rate policy cannot be characterized
reason is that a devaluation of the currency lowers the just by the cyclical properties of output. The source of
relative price of non-tradable goods, and this reduces the shock matters: while recessions induced by a drop in
the necessary adjustment in output relative to the case the price of exportable goods call for a devaluation of the
in which the exchange rate remains constant. This paper currency, those induced by a drop in productivity in the
uses a simple small open economy model with sticky non-tradable sector require a revaluation.
This paper--a product of the Growth and the Macroeconomics Team, Development Research Group--is part of a larger
effort in the department to understand the role of fiscal and monetary policy in developing countries. Policy Research Working
Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at chevia@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Optimal Devaluations
Constantino Hevia Juan Pablo Nicolini
The World Bank Universidad Di Tella
This paper started after a lunch with Eduardo Levy Yeyati and Ernesto Schargrodsky, and grew out
of several interesting discussions and conversations with Pedro Teles. We also would like to thank Andres
Neumeyer, Rodolfo Manuelli, Fernando Alvarez, Pat Kehoe and Raphael Bergoeing for comments. Finally,
we want to thank Charles Engel for a very clarifying discussion. But all errors are our own. The ...nd-
ings, interpretations, and conclusions expressed in this paper are entirely those of the author. They do not
necessarily represent the view of the World Bank, its Executive Directors, or the countries they represent.
1
INTRODUCTION
In most economies that can be described as small and open, the Central Bank targets the
value of the nominal exchange rate by intervening in the foreign currency market. Some-
times, the nominal exchange is ...xed at a given value for a period of time; most often, the
interventions respond to economic conditions in a discretionary way.
According to the conventional wisdom, when the economy enters a recession it is conve-
nient to devalue the domestic currency due to the existence of price stickiness. It is argued
that a devaluation lowers the relative price of non-tradable goodsthat would not fall oth-
erwise because nominal prices are stickyand, therefore, allows for a smaller adjustment in
quantities, making the recession less severe1 . This conventional wisdom has been supported
by the implications of static reduced form models in which the way ...rms that cannot adjust
prices behave, given that constraint, is not formally modeled. The policy implication derived
from these models is attractive in that it can be characterized in a simple way: the nominal
exchange rate must be counter-cyclical.
In this paper we characterize the optimal exchange rate policy in a fully speci...ed dynamic
general equilibrium model of a small open economy in which the price frictions are explicitly
modeled and ...rms have rational expectations. The model identi...es the distortion that price
stickiness introduces and the way this distortion interacts with other ones, like income taxes
and monopoly power. We can formally analyze how these di¤erent distortions should be
traded o¤ to maximize welfare, as it is traditionally done in the second-best literature. The
purpose of the paper is to explore if a standard general equilibrium dynamic Ramsey model
of a small-open economy supports the conventional wisdom.
We study a labor only representative agent economy with monopolistic competitionso
a
...rms have power to set prices cash-in-advance constraint, a ...nal non-tradable good, and
1
This was the logic behind several devaluations, like the ones in Brazil (1999) Argentina (2001) and, to
some extent, Spain, UK and other western European countries in the early 90´s.
2
two tradable intermediate inputs, one that can be exported and the other that must be
so
imported we can analyze the optimal policy response following both real exchange rate
and terms of trade shocks. Our model extends the closed economy model in Correia, Nicolini
and Teles (2008) to allow for international trade in goods and international capital mobility.
We show that the conventional wisdom (that is, that a devaluation must follow a recession)
does not hold even in the simple model we analyze: it depends on the source of the shock
generating the uctuation and on the modeling details. For example, if the recession is
due to a decline in the international price of exportable goods, the nominal exchange rate
should be devalued. On the other hand, if the recession is due to a negative productivity
shock in the non-tradable sector, the nominal exchange rate should be revalued. In addition,
modeling details such as the distinction between cash and credit goods or which is the sector
where the price friction is present also matter. For example, as we impose a cash-in-advance
constraint solely on consumers, it is never optimal to move the nominal exchange rate after
government spending shock, but an active exchange rate policy is called for after those same
shocks if the government also faces a cash-in-advance constraint.
We derive general principles for the conduct of optimal monetary policy which hold in
other contexts2 : optimal policy ought to fully stabilize the "right"price index and reproduce
the exible prices allocation. However, the way that principle translate into actual policy
instruments depends on both the source of the shock and on details of the economy such as
where exactly the rigidity is, which are the traded goods and which are the cash goods.
The policy instruments that we consider are labor income taxes, dividend taxes, a tax
on the return to foreign assets, and monetary policy. We also allow the government to
issue state contingent bonds. With these instruments, the government has a complete set of
instruments, as de...ned in Chari and Kehoe (1999). As it is standard in Ramsey analysis,
we abstract from time inconsistency and assume full commitment. Thus, whichever role the
2
See for instance Obstfeld and Rogo¤ (2000) and Woodford (2003).
3
exchange rate can have in fostering good (or bad) reputation will be absent in this analysis.
Following the seminal work of Obstfeld and Rogo¤ (1995, 1996) there has been a growing
interest in studying optimal policy in open economies with frictions in the setting of prices.
A branch of the literature, like Obstfeld and Rogo¤ (2000), Engel (2001) and many others3
focused on the two-countries case. This literature emphasizes the relationship between the
strategic interactions in two-countries models and optimal exchange rate policy, and in most
cases it focused on the exible versus ...xed exchange rate regimes debate. In addition, it
ignores - with the exception of Adao, Correia and Teles (2005) - the interaction between
exchange rate and monetary policy with distortionary ...scal policy, an issue that we formally
address.
The case of the small open economy we focus has been the subject of several papers. Gali
and Monacelli (2005) and de Paoli (2004) study a similar problem but do not consider the
...scal instruments as we do. Their approach only allows solving for linear approximations
very useful to compare alternative rules, but less so to characterize properties of the optimal
policy as we do. Cunha (2001) analyzes a model similar to ours but without a complete set
of instruments and he can only solve some examples. Closer to ours is the work of Faia and
Monacelli (2004a) and (2004b), who use a framework similar to ours, but without jointly
considering ...scal and monetary instruments.
The main contribution of this paper is the focus: rather than studying the welfare prop-
erties of families of policy rules in general, we want to answer the following question: should
a government of a small open economy devalue during a recession? The key di¤erence with
the literature is the joint analysis of monetary and ...scal policy with distortionary taxation.
It turns out that with a complete set of instruments, the analysis of the solution is rather
simple since all known results on the Ramsey literature can be applied. Therefore, we can
3
An incomplete list also includes Corsetti and Pesenti (2001) and (2005), Devereux and Engel (2003),
Benigno and Benigno (2003), Duarte and Obstfeld (2004), Ferrero (2005) and Adao, Correia and Teles
(2005).
4
provide a very sharp answer to the question we focus on.
The paper is organized as follows. Section 2 spells out the model. Section 3 solves for
the optimal allocation and characterizes the optimal monetary and exchange rate policy as
a function of the shocks to the economy. Section 4 discusses extensions and concludes.
1. THE MODEL
We model a dynamic small open economy with uncertainty. Time is discrete and the state
of the economy at period t = 0; 1; ::: is denoted by st , which belongs to a ...nite set of events,
with s0 given. Let st = (s0 ; s1 ; :::; st ) be the history of events up to period t, and let (st )
be the probability of st conditional on s0 .
Households: There is a representative household who derives utility from a composite
good and leisure according to
XX
1
t
U= U c st ; ` st st (1)
t=0 st
where 0 < < 1; U ( ) is increasing in each argument and concave, ` (st ) denotes leisure,
and c (st ) is a composite ...nal goods de...ned as
Z 1 1 1
t
c s = c i; st di :
0
The variable c (i; st ) denotes consumption of a ...nal good of variety i 2 [0; 1], and > 1 is
the elasticity of substitution across varieties.4 We assume that all varieties are non-traded,
so they must be produced domestically.
Markets are complete. We let B (st ; st+1 ) and B (st ; st+1 ) be one-period bonds denomi-
nated in domestic and foreign currency respectively. These bonds are issued at st and pay
4
Existence of an equilibrium requires > 1.
5
one unit of the corresponding currency at t + 1 on the event st+1 and zero otherwise. The
price in domestic and foreign currency of these claims are Q (st+1 jst ) and Q (st+1 jst ) respec-
tively. For k > 0, let Q st+k jst = Q (st+1 jst ) Q (st+2 jst+1 ) :::Q st+k jst+k 1
be the price
of one unit of currency at st+k in units of currency as st . A similar de...nition holds for
Q st+k jst : As long as a no-arbitrage condition between domestic and foreign bonds is sat-
is...ed (see condition (18) below), we can assume, without loss of generality, that households
do not hold foreign bonds.5 Therefore, for all periods, households face the budget constraint
X Z 1
h t t+1 t+1 t h t 1
M s + B s Q s js M s Pt 1 i; st 1
c i; st 1
di (2)
st+1 0
+W st 1
1 ` st 1
1 n
st 1
+ B st ;
where W (st ) (1 n
(st 1 )) is the net of taxes nominal wage rate, M h (st ) is the household'
s
demand for currency, and Pt (i; st ) is the price of the ...nal good of variety i. We assume that
initial nominal wealth is zero: M 1 + B0 = 0.6
We model a money demand by imposing a cash-in-advance constraint on a subset of goods.
The timing protocol follows Lucas (1984) and we assume that goods in the set [0; 1] are
cash goods, so we impose the following cash-in-advance constraint
Z
P i; st c i; st di M h st : (3)
By making a proper subset of the unit interval, we allow for cash and credit goods in the
model.
5
Note that markets are complete with the domestic securities, so the foreign currency denominated secu-
rities are redundant. It is convenient to introduce these bonds for two reasons, ...rst, to let all intertemporal
trade with the rest of the world be carried out by issuing foreign securities, and second, to derive the interest
parity conditions.
6
We assume this in order to avoid the well known problem of hyperination incentives in period 0 (Lucas
and Stokey 1983).
6
If we use the compact notation x to denote the contingent sequence fx (st )g for any x, the
household' problem is to maximize (1), by choice of c (i) ; `; M h ; B , subject to (2), (3), and
s
an arbitrarily large negative lower bound on the real holding of assets. If the cash-in-advance
constraint is binding, the optimality conditions can be expressed as
U` (st ) W (st ) n
= 1 st (4)
Uc (st ) Pt (st )
Uc (st+1 ) Pt (st ) R (st+1 )
Q st+1 jst = st+1 jst (5)
Uc (st ) Pt+1 (st+1 ) R (st )
Pt (st )
c i; st = c st if i 2 (6)
R (st ) Pt (i; st )
Pt (st )
c i; st = c st if i 2 c
(7)
Pt (i; st )
where (st+1 jst ) is the conditional probability of st+1 given st , and Pt (st ) is a price index
for households de...ned as
Z Z 1
1
t t 1 t 1 t 1
Pt s R s Pt i; s di + Pt i; s di :
c
Here Uc (st ) and U` (st ) denote the marginal utility of consumption and leisure at st respec-
hP i 1
t t+1 t
tively, R (s ) = st+1 Q (s js ) is the nominal (gross) interest rate between periods
t and t + 1, and Pt (st ) is the price of a unit of consumption aggregate c (st ).7 The price
index incorporates the fact that the e¤ective price of a cash good, Pt (i; st ) R (st ), includes
the ...nancial cost of holding currency. Equation (4) is the standard intratemporal condition
between leisure and the consumption aggregate; (5) is the intertemporal optimality condi-
tion; and (6) and (7) are the conditional demands of cash and credit goods, respectively.
7
It is standard to show that if R (st ) > 1 the cash-in-advance constraint binds, while if R (st ) = 1 it does
not. In this latter case we will focus on the equilibria where (3) holds at equality.
7
Further, (6) and (7) can be expressed as
1=
c (i; st ) Pt (j; st ) c
= if i 2 and j 2 : (8)
c (j; st ) R (st ) Pt (i; st )
This condition equates the marginal rate of substitution between cash and credit goods to
their relative price.
Government: The government has to ...nance an exogenous sequence of expenditures,
which is also a composite of non-tradeable goods identical to that of the households,
Z 1 1 1
g (st ) = g i; st di ;
0
where g (i; st ) is the government purchase of the ...nal good of variety i. Taking as given the
prices Pt (i; st ) ; the government minimizes the cost of acquiring the aggregate g (st ) : Cost
minimization implies the conditional demands
Ptg (st )
g i; st = g (st ) . (9)
Pt (i; st )
hR i11
1 t 1
where Ptg t
(s ) = 0
Pt (i; s ) is the cost minimizing price index for the government.
Households and the government face di¤erent price indexes because the former are subject
to a cash-in-advance constraint.
Monetary policy consists of rules for the stock of money M (st ), and the nominal exchange
rate between domestic and foreign currency E (st ). Fiscal policy consists of linear tax rates
n
on labor (st ); taxes on dividends d
(st ); linear taxes on foreign securities (st ); one-
period bonds issued to domestic residents denominated in domestic currency B (st ; st+1 ) ;
and one-period state contingent bonds issued to foreigners denominated in foreign currency
B (st ; st+1 ). A government policy is de...ned as ! M; E; n
; d
; ; B; B , where we use
8
the compact notation described above.
The introduction of taxes on foreign securities is useful for two reasons: ...rst, it roughly
captures the idea of taxing international capital ows, and second, they are used to implement
the Ramsey allocation. In the last section we discuss the consequences of eliminating these
taxes, and how they can be substituted by consumption taxes. In addition, since taxation
of dividends is equivalent to lump-sum taxes, any optimal policy will set dividend taxes to
d
its highest feasible value (st ) = 1 for all st .8
Tradeable Sector: The non-tradeable varieties are produced using two tradeable inputs.
One of them can be produced domestically and the other must be imported.9 Output of the
home - exportable - input is given by the linear production function
X st = A (st ) n st
where A (st ) is a technology shock and n (st ) is labor. Competitive pricing requires
Px st = W st =A (st ) : (10)
where Px (st ) is the domestic currency price of the home input.
As this is a small open economy, we assume that international trade is carried out in
foreign currency. Absence of arbitrage opportunities implies the purchasing power parity
conditions
Px s t = E st Px (st ) (11)
Pm st = E st Pm (st ) :
8
The results also hold if d (st ) = 0 or if it is constrained to be a positive number lower than one.
The problem is much simpler with our assumption, otherwise the governemnt will try to manipulate other
instruments, like monetary policy or exchange rates in order to tax pro...ts.
9
This allows consideration for terms of trade shocks.
9
where Pm (st ) is the domestic currency price of the foreign input and the terms of trade are
de...ned as d (st ) = Px (st ) =Pm (st ) for all t:
Production of Varieties: The technology to produce the ...nal good i 2 [0; 1] is given by
the Cobb-Douglas production function
1
y i; st = Z (st ) x i; st m i; st
where x (i; st ) is the home input, m (i; st ) the foreign input, and Z (st ) is an aggregate
productivity shock common across varieties.10
The Cobb-Douglas technology implies that marginal cost is given by
1
Px (st ) Pm (st )
M C st = :
Z (st ) (1 )1
Using (11) and the de...nition of d (st ) ; the marginal cost function can be written, in equi-
librium, as
Pm (st ) d (st )
M C st =
(1 )1 Z (st )
Cost minimization implies that all ...nal goods ...rms choose the same inputs ratio
m (i; st ) 1
= d (st ) for all i: (12)
x (i; st )
Then, equilibrium production can be expressed as
Z (st )
y i; st = m i; st : (13)
d (st ) 1
10
Our results generalize to any constant returns to scale technology y (i; st ) = F (x (i; st ) ; m (i; st )) but
at the cost of additional notation, see Hevia and Nicolini (2004).
10
We assume that each variety is produced by a monopolist.11 We also assume that there
is price stickiness. In particular, ...rms i 2 [0; ] are constrained to set prices at period t
conditional on the information available up to period t 1. Equivalently, we can think of
those ...rms, called sticky ...rms, as setting prices one period in advance.12 The other ...rms,
called exible ...rms, are allowed to set prices at t conditional on st .
The demand faced by ...rm i is y d (i; st ) = c (i; st ) + g (i; st ), and di¤ers according to the
cash-credit characteristic of the good. Equations (6), (7) and (9) imply
y d i; st = Pt i; st e
Y i; st ;
where 8
>
< c (st ) [Pt (st ) =R (st )] + g (st ) P g (st )
t if i 2
e
Y i; st = :
>
: t t
c (s ) Pt (s ) + g (st ) Ptg t
(s ) if i 2 c
Note that both the marginal cost and the elasticity of demand that each monopolist face
is the same across varieties. Thus, from the point of view of preferences and technologies,
the model exhibits total symmetry. However, the frictions in transactions and in the setting
of prices do introduce heterogeneity across ...rms. Indeed, there are four types of ...nal goods
...rms: exible ...rms with and without cash-in-advance constraint, and sticky ...rms with and
without cash-in-advance constraints.
Flexible ...rms face the static optimization problem of maximizing nominal dividends,13
max Pt i; st M C st y d i; st :
Pt (i;st )
11
This assumption is essential to have ...rms that can set prices in advance.
12
The results also hold for other forms of price stickyness, as Calvo staggered pricing for instance. For
details, see Correia, Nicolini and Teles (2002).
13
Strictly speaking, full taxation of dividends imply that the pricing and production decisions of the ...rms
are indeterminate. It is convenient, instead, to think of each ...rm maximizing after-tax dividends for d < 1,
j
and then considering the limit economy as limj!1 d = 1 for all st .
j
11
Independently of whether the good is cash or credit, the optimal pricing rule determines the
price as a constant mark-up over the marginal cost,
Pm (st ) d (st )
Pt i; st = for i 2 ( ; 1]: (14)
1 (1 )1 Z (st )
Sticky ...rms set prices at period t conditional on information available up to period t 1.
This is equivalent to choosing prices at t s
1 to maximize the value of the next period'
dividends,14
X
max1
t
Q st jst 1
Pt i; st 1
M C st y d i; st
Pt (i;s )
st jst 1
The optimal pricing rule is
X Pm (st ) d (st )
Pt i; st 1
= i; st for i 2 [0; ] ; (15)
1 st jst 1
(1 )1 Z (st )
where
e
Q (st jst 1 ) Y (i; st )
t
i; s =P :
e
t t 1 ) Y (i; st )
t t 1 Q (s js
s js
The optimal price is set as a mark-up over a weighted average of the marginal cost across
states. Unless the government follows the Friedman rule ( i.e. R (st ) = 1), sticky ...rms will
choose di¤erent prices depending on whether the good they produce is cash or credit.
In addition, we assume throughout that the initial price at period 0 of all sticky ...rms are
s
identical and given by P0 .
Foreign Sector: The trade balance measured in units of the home input is de...ned as
Z 1 Z 1
t t t Pm (st )
TB s =X s x i; s di m i; st di; (16)
0 Px (st ) 0
14
We assume that shocks are su¢ ciently small so that sticky ...rms always ...nd it optimal to remain active.
12
R1 Pm (st ) R1
where X (st ) 0
x (i; st ) di is net exports of the home input and Px (st ) 0
m (i; st ) di denotes
the imports of the foreign input measured in units of the home input.
s
The evolution of the country' foreign debt is given by
X
B st+1 Q st+1 jst + Px (st 1 ) T B st 1
=B st
st+1 jst
where B (st ) denotes the stock of foreign debt of the economy as a whole and T B (s 1 ) = 0.
Solving the previous equation starting from period 0 forward and ruling out Ponzi schemes,
we obtain the economy foreign sector feasibility constraint,
XX
1
Q (st js0 )
Px (st ) T B st = B0
R (st )
t=0 st
1
where B0 is initial debt and R (st ) = [Q (st+1 jst )] is the foreign interest rate.
Foreign investors are risk neutral and discount the future at the same rate as domestic
residents, hence the price Q satis...es
Px (st ) R (st+1 )
Q st+1 jst = st+1 jst
Px (st+1 ) R (st )
Thus, the foreign sector constraint becomes
XX
1
B0 R (s0 )
t
T B st st = (17)
P (s0 )
t=0 st
Finally, by arbitrage, the price of domestic and foreign bonds must be related through the
covered interest parity condition
E (st )
Q st+1 jst = Q st+1 jst 1+ st+1 : (18)
E (st+1 )
13
Market Clearing: Feasibility in the ...nal goods, labor and currency market require
c i; st + g i; st = y i; st , for i 2 [0; 1] ; (19)
n st = 1 ` st (20)
M h st = M st : (21)
De...nition: An allocation a and a price system P are contingent sequences a = fc (i), `,
x (i), m (i), y (i)g and P = fQ, R, Px , Pm , P (i), W g for i 2 [0; 1].
De...nition: Given a government policy !, an allocation a and a price system P are an
equilibrium if (i) households solve their utility maximization problem; (ii) the price of the
home input satis...es (10); (iii) ...nal goods producers act optimally: (12) hold for all i 2 [0; 1]
and they follow the pricing rules (14) if i 2 ( ; 1] and (15) if i 2 [0; ]; (iv) the market clearing
conditions (19), (20) and (21) are satis...ed; (v) the economy-wide feasibility constraint (17)
holds; (vi) the no-arbitrage conditions (11) and (18) hold; and vii) net nominal interest rates
are non-negative, R (st ) 1 for all st . (By Walras'Law, the government budget constraint
is also satis...ed.)
2. THE RAMSEY PROBLEM
As it is standard in the literature, we assume the government can commit to a particular
policy chosen at period 0. We de...ne an allocation rule a (!) as the mapping from the set
of government policies into equilibrium allocations. Speci...cally, a (!) is the set of allocation
that satisfy conditions i) to vii) in de...nition 2 given !, and we call these allocations imple-
mentable. The Ramsey problem is to choose the implementable allocation that maximizes
s
the household' utility (1). Let the solution, called the Ramsey allocation, be denoted by
aR .
14
The standard approach (Lucas and Stokey (1983), Chari and Kehoe (1999)) is to ...nd
necessary and su¢ cient conditions that an allocation has to satisfy to be implementable.
Let the set of allocations that satisfy the necessary and su¢ cient conditions, called the
implementable set, be denoted by J ( ; ).15 Then, aR is de...ned as
aR 2 arg max U (a) :
a2J( ; )
As in this model the su¢ cient conditions cannot be characterized in terms of the allocations
alone, we follow a di¤erent approach to characterize the Ramsey allocation. We ...rst describe
a set of necessary conditions any equilibrium allocation must satisfy.
Proposition 1: Given a government policy !, if an equilibrium exists, the allocation a (!)
satis...es the following conditions:
i) feasibility:
Z (st )
c i; st + g i; st = m i; st ; (22)
d (st ) 1
ii) current account sustainability:
XX
1
B0 R (s0 )
t
T B st st = ; (23)
P (s0 )
t=0 st
where R1
m (i; st ) di
T B st = A (st ) 1 ` st 0
;
(1 ) d (st )
and
s
iii) household' optimization
XX
1
t
Uc s t c s t U` st 1 ` st st = 0: (24)
t=0 st
15
Given a policy, the equilibrium allocation in general depends on the parameters and :
15
Proof: Condition i) follows from (13) and (19). To obtain condition ii), use the de...nition
of X (st ) and (20) into (16), and note that (11) and (12) imply
Z Z R1
1
t Pm (st ) 1
t 0
m (i; st ) di
x i; s di + m i; s di = :
0 Px (st ) 0 (1 ) d (st )
s
Finally, from (2) construct the household' present value budget constraint
X X Q (st js0 )
1 Z 1
Pt i; st c i; st + M (st ) (R (st ) 1) W st 1 n
st 1 ` st =0
R (st ) 0
t=0 st
where we assume B0 + M 1 = 0. Condition (24) follows, as it is standard, by using (3), (4)
and (5) into the previous equation.
Conditions (22) and (23) are feasibility in the ...nal goods market and foreign-sector feasi-
s
bility, and (24) summarizes the household' behavior.
It is important to stress that (22), (23) and (24) are not su¢ cient conditions of an equi-
librium allocation. In addition to those conditions, an equilibrium allocation must satisfy a
symmetry constraint across ...nal goods ...rms of each type. There are four types of ...nal goods
...rms, the four possible combinations of sticky-exible, and cash-credit goods. In equilibrium,
...rms of each type choose the same prices and inputs. These constraints imposed by the price
setting and ...nancial restrictions make the equilibrium set J ( ; ) a complicated object to
e
work with. Thus, we ...nd it convenient to focus on a larger set J de...ned as follows,
e
De...nition: Let J be the set of allocations satisfying the conditions in proposition 1.
e
As the conditions in proposition 1 are necessary, it is clear that J includes the equilibrium
set J ( ; ) for any pair ( ; ).16 Our strategy consists on ...nding the best allocation in the
e
set J; and then showing that the argmax belongs to the set J ( ; ) for any pair ( ; ) : In
other words, since the equilibrium set J ( ; ) is a complicated object, we ...nd the allocation
16 e
It is easy to see that the set J is independent of both and :
16
s e e
that maximizes the household' utility in the set J. Since J includes J ( ; ), we are solving
a relaxed Ramsey problem which maximizer allocation may not be an equilibrium. We
then show, however, that for all ( ; ) there is a government policy ! R and a price system
P R such that the relaxed Ramsey allocation satis...es the additional equilibrium conditions.
Consequently, the relaxed Ramsey allocation can be implemented as an equilibrium and
e
therefore, it is the Ramsey allocation. Interestingly, since the larger set J is independent of
the degree of price stickiness , and of the set of cash-goods , it follows that the Ramsey
allocation aR is also independent of and .17
The relaxed Ramsey allocation e is de...ned as
a
e = arg max U (a) :
a
e
a2J
Since J ( ; ) e
J; it follows that U (e)
a U aR . Proposition 2 below states that, in fact,
e = aR ; so U (e) = U aR .
a a
Relaxed Ramsey Problem: The relaxed Ramsey problem is to maximize (1), by choice
of fc (i) ; `; m (i)g, subject to the necessary conditions (22), (23), and (24). We solve this
problem by posing the Lagrangian
XX
t
L = st U c st ; ` st + Uc st c st U` s t 1 ` st
t st
Z 1
Z (st )
c i; st + g i; st
i; st m i; st di
0 d (st ) 1
" R1 #)
t
m (i; s ) di
+ A (st ) 1 ` st 0
:18
(1 ) d (st )
t
where (st ) (i; st ) ; and are the Lagrange multipliers on (22), (23) and (24), respec-
tively.
17
For a further discussion of this result in a closed economy model, see Correa, Nicolini and Teles (2002).
17
Lemma 1: The relaxed Ramsey allocation satis...es c (i; st ) = c (st ) for all i and st . In
addition, if e is implementable, then g (i; st ) = g (st ) ; x (i; st ) = x (st ) and m (i; st ) = m (st )
a
for all i and st : Furthermore, prices satisfy Pt (i; st ) = Pt (st ) for all i and st ; and the
Friedman rule is optimal, that is R (st ) = 1 for all st :
Proof: Taking the necessary ...rst order conditions of the Lagrangian with respect to
m (i; st ) and c (i; st ) ; we obtain
1
i; st = 1
(1 ) Z (st ) d (st )1
1=
t t t t t c (i; st )
Uc s (1 + ) + Ucc s c s U`c s 1 ` s = i; st
c (st )
The last two equations and the de...nition of c (st ) imply c (i; st ) = c (st ) for all i and st . It
follows from the de...nition of Pt (st ) and conditions (6) and (7) that Pt (st ) = Pt (i; st ) for all i
and st . Then (8) and the previous results imply R (st ) = 1 for all st : Further, g (i; st ) = g (st )
for all i and st follows from (9), and equations (22) and (12) imply m (i; st ) = m (st ) and
x (i; st ) = x (st ) for all i and st .
Corollary: If > 0; all prices at period t depend on st 1 . That is, the price of the ...nal
goods are perfectly forecastable one period in advance.
e
The intuition is as follows: the set J does not restrict the behavior of the ...nal goods
...rms. Since any pair of ...nal goods c (i; st ) and c (j; st ) for i 6= j are imperfect substitutes in
the homogeneous consumption aggregate c (st ), it is optimal to set a symmetric allocation
in which c (i; st ) = c (st ) for all i. If implementable, a symmetric allocation requires an
equal price across varieties. The prices that matter are those faced by the households and
the government, and incorporate the ...nancial cost of holding currency in order to buy
cash-goods. Therefore, this requires an identical nominal price P (i; st ) across varieties and
18
following the Friedman rule R (st ) = 1. In addition, if > 0; a fraction of ...rms set prices
at t; conditional on the information at t 1, hence equal nominal prices requires the prices
of all varieties to depend on information available up to period t 1.
The next proposition shows that indeed, the relaxed Ramsey allocation e is implementable
a
as an equilibrium for any degree of price stickiness .
Proposition 2: For any pair ( ; ) there is a government policy ! R and a price system
P R that implement the relaxed Ramsey allocation e. Therefore, e = aR . Proof: in the
a a
appendix.
distorted'
It is interesting to note that the relaxed Ramsey problem solves a ...rst best for a `
utility function de...ned as
e
U c st ; ` st ; U c st ; ` st + Uc st c st U` st 1 ` st :
e
If U (c (st ) ; ` (st ) ; ) is well behaved (i.e. increasing and concave),19 the qualitative response
of the Ramsey allocation to the di¤erent shocks is identical to the response in the ...rst best,
where the planner maximizes (1) subject to the feasibility constraints.
e
Using that the allocation is symmetric and the de...nition of U (c; `; ) ; the necessary ...rst
order conditions of the Ramsey problem can be expressed as
e 1
Uc c st ; ` st ; = 1 (25)
(1 ) Z (st ) d (st )1
e
U` c st ; ` st ; = A (st ) (26)
19
This may not be the case, since Uc (st ) c (st ) may not be concave in consumption.
19
Feasibility in the ...nal goods market becomes
Z (st )
c st + g (st ) = m st : (27)
d (st ) 1
Equations (25), (26) and (27) together with constraints (23) and (24) characterize the relaxed
Ramsey allocation, where the trade balance becomes
m (st )
T B st = A (st ) 1 ` st : (28)
(1 ) d (st )
Speci...cally, equations (25) and (26) can be used to solve for consumption and leisure as a
function of the multipliers and . Then we can use (12) and (27) to solve for m (st ) and
x (st ) as a function of and . The present value constraints (23) and (24) can be used to
...nd the values for and .
Equations (25), (26), (27) and (28) imply that the Ramsey allocation only depends on
the realization of the stochastic processes dated at period t, as government purchases g (st ),
productivity shocks A (st ) and Z (st ), and the terms of trade shock d (st ), and not on the
history of realizations st . In addition, government expenditure shocks do not a¤ect c (st )
nor ` (st ) since both can be solved as a function of the shocks A (st ), Z (st ) and d (st ),
and of the two multipliers and , which do not depend on the particular realization of
g (st ). For example, if government expenditures increase, equation (27) shows that in order
to keep private consumption constant, the usage of the tradeable inputs m (st ) and x (st )
increase. Then, (28) shows that the trade balance decreases. In other words, the availability
of international credit allows the planner to insure all government expenditure shocks through
borrowing and lending.
In what follows we assume that utility is separable between consumption and leisure and
20
e
that is well behaved, so U (c; `) = H (c; ) V (1 `; ) is increasing and concave.20 We will
analyze these cases ...rst, and we discuss other cases in the last section.
Response to Shocks: Here we study how the di¤erent shocks a¤ect the optimal alloca-
tion. Suppose that we want to analyze a shock to the ...nal good technology Z (st ). Strictly
speaking, given st 1 ; the analysis compares two states st and s0t such that Z (st ) > Z (s0t ) :
However, given the stationarity of the Ramsey allocation in our set-up, this is equivalent to
analyze the experiment Z (st ) > Z (st 1 ). Thus, both interpretations follow from our analy-
sis. In addition, since government expenditure shocks were analyzed above, here we focus
on productivity shocks Z (st ) and A (st ) ; and terms of trade shocks d (st ).
1) Consider a negative shock to the ...nal goods technology Z (st ). Leisure remains constant,
consumption decreases. The change in the trade balance and the change in usage of inputs
m (st ) and x (st ) are indeterminate.
Proof: Conditions (25) and (26) imply that c (st ) decreases and ` (st ) remains constant.
From (27) it follows that x (st ) could either increase or decrease. Hence from (12) and (28),
m (st ) and T B (st ) could increase or decrease.
The constancy of labor supply is a consequence of the separable utility function and the
fact that the country has access to international capital markets. This can be seen in equation
(26), where it is evident that the allocation of labor depends solely on the shocks to the home
input sector and the multipliers, which capture the wealth of the country in present value
terms. The usage of inputs x (st ) and m (st ) could increase or decrease depending on the
curvature (i.e. the willingness to substitute intertemporally) of the utility function.
2) Consider a negative shock to the terms of trade d (st ) : Leisure remains constant and
consumption decreases. The usage of the foreign input m (st ) decreases. The change in x (st )
and T B (st ) are indeterminate.
20
In this case, charaterizing the reaction of the optimal allocation after a shock is equivalent to a ...rst
best and the intuitions are straighforward.
21
Proof: (25) and (26) imply that c (st ) decreases and ` (st ) does not change. From (27) it
follows that m (st ) decreases. Then (12) imply that x (st ) could increase or decrease. Finally,
from (28), T B (st ) could increase or decrease.
Intuitively, a negative shock to the terms of trade has, in addition to a negative income
e¤ect, a substitution e¤ect. The country is poorer since the price of the exportable input
is lower. The negative income e¤ect implies a reduction in consumption. Labor remains
constant for the same reason as in 1) above. The income e¤ect implies a lower usage of the
inputs x (st ) and m (st ) to produce less ...nal goods. But the reduction in the relative price
of x (st ) relative to m (st ) implies a substitution toward the home input. Both e¤ects imply
an reduction in m (st ) ; but the net e¤ect in x (st ) and the trade balance, are indeterminate.
3) Consider a negative shock to the home input technology A (st ) : c (st ) ; m (st ) and x (st )
do not change. Leisure increases and the trade balance decreases.
Proof: Equations (12), (25) and (27) imply that neither c (st ) ; m (st ) ; nor x (st ) change.
Condition (26) implies that ` (st ) increases, and hence, from (28), T B (st ) also decreases.
The shock to the home-input technology determines when it is good a time to export and
when it is not. Periods with low values of A (st ) are bad periods to export, hence leisure
increases and the trade balance decreases.
The table below summarizes these results
c (st ) ` (st ) x (st ) m (st ) T B (st )
# Z (st ) # = = # #
# d (st ) # = = ? ?
# A (st ) = " " = #
Table I. Response of the allocation to negative shocks
Note that for any shock, either consumption or labor remains constant. This is because of
the joint e¤ect of two assumptions: separability of labor and consumption in preferences and
22
the fact that labor does not enter the production function of the varieties. How changing
any of these assumptions a¤ect the results will be discussed in the last section.
3. DECENTRALIZATION
This section studies the policy implications regarding monetary and exchange rate policy
in a partially sticky economy ( i.e. 0 < < 1 ) with separable utility. The complete
characterization of the optimal policy for any 2 [0; 1] and any utility function is in the
appendix.
As mentioned above, the Friedman rule is optimal since it eliminates the distortion between
cash and credit goods, then R (st ) = 1 for all st . Given the Ramsey allocation aR and the
s
initial price P0 of sticky ...rms, we consider the following equilibrium conditions
H 0 (c (st ) ; ) X H 0 (c (st+1 ) ; )
= st+1 jst ; (29)
Pt (st 1 ) t+1 t
Pt+1 (st )
s js
which is the Fisher equation, obtained by summing (5) for all st+1 jst and setting R (st ) = 1
for all st ,
M s t = Pt s t 1
c (st ) (30)
R
is the cash-in-advance constraint, where = size'of the set ;
di is the `
1 Z (st )
Pm s t = (1 )1 Pt s t 1
(31)
d (st )
is the pricing rule of exible ...rms once we impose that it must be equal to the price of the
sticky ...rms, and
V 0 (1 ` (st ) ; ) 1
= (1 )1 Z (st ) A (st ) d (st )1 1 n
st ; (32)
H 0 (c (st ) ; )
23
s
the household' equilibrium condition (4) after using (10), (11) and (31).
The equilibrium policies and prices are obtained as follows. Given the initial price P0 (s 1 ) =
s
P0 , (29) determines recursively the price of the ...nal good. The cash-in-advance constraint
(30) determines the money supply at period t that implements the price Pt (st 1 ). Condition
(31) determines the foreign-input price Pm (st ) and (32) pins down the labor tax rate n
(st ).
The exchange rate and home-input price follow from (11), and (10) determines the nominal
wage rate W (st ) : The prices Q (st+1 jst ) follow from (5), and the taxes on foreign securities
(st+1 ) are pinned down from (18). (The decentralization of asset holdings is described in
the appendix.)
Policy Response to Shocks: In all cases, the behavior of labor taxes is indeterminate,
except for a government purchase shock, where they stay constant. Moreover, note that
none of the instruments or prices change with government expenditure shocks. This is a
consequence of the previously mentioned result that expenditure shocks are fully insured
through international capital markets. We study the optimal response of policy after the
other three shocks.
1) Negative shock to the ...nal goods technology Z (st ). Consumption decreases, and labor
remains constant. Since Pt (st 1 ) is given, it follows from (30) that a lower consumption is
implemented through a reduction in M (st ) : In addition, the input prices, the nominal wage
rate and the exchange rate all decrease. The decline in Pm (st ) ; Px (st ) and E (st ) follow
from (31) and (11), and the decrease in W (st ) follows from (10).
2) Negative shock to the terms of trade d (st ) : Consumption decreases, labor remains con-
stant and m (st ) decreases. The decentralization of a terms of trade shock di¤ers depending
on whether the decline in d (st ) is driven by a lower Px (st ) or a higher Pm (st ). In any
case, from (30), the lower consumption level is implemented through a decline in M (st ).
24
Condition (31) implies that Pm (st ) increases but d (st ) Pm (st ) does not change.21 Since
Px (st ) = d (st ) Pm (st ), the home-input price does not change, and therefore, (10) implies
that W (st ) also remains constant. If the negative shock to d (st ) is driven by a decline
in Px (st ) ; (11) implies that E (st ) increases. If the decline in d (st ) is driven solely by an
increase in Pm (st ) ; from (11) and (31), the nominal exchange rate does not change. (Due to
this result, the next table considers changes in Px (st ) and Pm (st ) separately.)
3) Negative shock to the home-input technology A (st ). Consumption remains constant and
labor decreases. (30) implies that M (st ) does not change. (31) implies that Pm (st ), and
from the (11) conditions, Px (st ) and E (st ) do not change. Finally, condition (10) implies
that W (st ) declines.
The following table summarizes the previous results
E (st ) M (st ) W (st ) Pm (st ) Px (st )
# Z (st ) # # # # #
# Px (st ) " # = " =
" Pm (st ) = # = " =
# A (st ) = = # = =
Table II. Response of the optimal policies and prices to negative shocks
Table II summarizes the main message of the paper: it is not possible to characterize
optimal policy by the cyclical properties of the economy even in this very simple model. The
table considers several negative exogenous shocks hitting the economy. Some shocks call for
a depreciation of the exchange rate, others call for an appreciation of the exchange rate, and
others call for a constant exchange rate.
21
The last result is special to the Cobb-Douglas speci...cation.
25
4. EXTENSIONS AND CONCLUSIONS
In this section we discuss extensions to the model we just analyzed and show that they
reinforce the results obtained. We assumed that labor enters only in the production of the
tradeable input and that consumption and labor were separable in preferences. These two
assumptions imply that either leisure or consumption would remain constant after any of
the shocks analyzed and made the discussion of the decentralization straightforward and
simple. The model can be extended by relaxing any of these assumptions. For instance, if
we introduce labor as an additional input in the production of the continuum of ...nal goods,
a negative shock to the home-input technology A (st ) can be shown to imply a devaluation
of the exchange rate and an increase in the money supply.22 These results contrasts with
those in Table II, where the nominal exchange rate and the money supply do not change.
We draw two conclusions from the discussion. First, that modelling details are important
for the qualitative predictions of the model, and second, that the main message that it is
not possible to characterize optimal policy in terms of simple rules based on the cyclical
properties of the economy is a robust result.
Another key assumption in the model is the existence of taxes on foreign securities (st ) :
This instrument gives the planner the ability to manipulate the intertemporal prices faced
by the domestic residents without violating the interest parity condition (18). The same
allocation can be implemented if, instead of having the taxes (st ), the government has
access to a uniform consumption tax over varieties i 2 [0; 1] : If, however, the government
does not have access to consumption taxes, the constraint (18) binds. The allocation will be
di¤erent, and welfare will be lower. Further, even though the consumption aggregate c (st ) is
homogeneous of degree one in the individual varieties, a condition known to imply that the
Friedman rule is optimal with a cash-in-advance constraint on a subset of varieties (Chari,
22
This version of the model is analyzed in Hevia and Nicolini (2004).
26
Christiano and Kehoe, 1991), it will not be so in this case, since the government does not have
a complete set of instruments. Intuitively, there is a trade-o¤ between distorting the relative
consumption of cash and credit varieties, and the constraint imposed by (18), which ties
down the intertemporal prices faced by households. By manipulating the nominal interest
rate the government a¤ects the intertemporal prices, but distorts the relative consumption
of varieties.23 The same happens if the tax rates cannot be made state contingent or if the
government cannot issue state contingent debt.
We imposed frictions in the setting of ...nal good prices. An alternative, widely used
in the literature, is to assume frictions in the setting of wages. In addition, we imposed
a one period rigidity, while most of the literature assumes staggered price setting. If we
relax both assumptions, the Ramsey allocation is still independent of both the set of cash
goods and the set of ...rms with sticky wages, and the same as in our model. However, the
decentralization would be very di¤erent, since monetary and exchange rate policy would aim
at stabilizing nominal wages, rather than prices. So, the optimal response of the nominal
exchange rate-- to implement the same Ramsey allocation-- would be di¤erent in general.
Finally, if the conditions imposed on the utility function in Section 3 are not satis...ed,
some of the conclusions regarding the optimal response of the allocation may be di¤erent.
This is a standard feature of optimal taxation Ramsey problems and means that some of
the results described in Table I may be di¤erent from what we obtained. However, given
a particular movement of the optimal allocation, the way to decentralize it is the same as
before.
This discussion reinforces the message of the paper: optimal policy is about dealing with
distortions in response to shocks. The optimal exchange rate policy critically depends on
which is the set of instruments available, on how -given those instruments- the optimal
allocation responds to the shocks and on the mapping from policies to allocations.
23
These results are discussed in Hevia and Nicolini (2004).
27
There is a general policy principle that can be derived from our analysis, which is con-
sistent with previous work: price stability is optimal, since this price stability is the one
that avoids the distortion between exible and sticky varieties. But to make this principle
operational, one needs to know, ...rst, where the price stickiness is, and second, which is the
transmission mechanism from the exchange rate to those sticky prices. The qualitative rela-
tionship between movements in aggregate output and optimal movements on the exchange
rate critically depends on those modeling details.
28
Appendix.
Proof of Proposition 2: We ...nd a government policy ! and a price system P such that
the relaxed Ramsey allocation satis...es conditions i) through vii) in de...nition 2. Throughout,
...x the relaxed Ramsey allocation e. Recall that e satis...es c (i; st ) = c (st ) and m (i; st ) =
a a
t t
x (s ) : As shown above, this requires P (i; s ) = P (st ) for all i, and R (st ) = 1 for all st .
P
Summing (5) over all st+1 jst and using st+1 jst Q (st+1 jst ) = 1, it follows that
Pt (st ) X Uc (st+1 )
1= st+1 jst (A1)
Uc (st ) t+1 t Pt+1 (st+1 )
s js
Using (21) into (3) we obtain the equilibrium condition,
M st = Pt st c st ; (A2)
R
where = di is the measure of the set : Solving the households budget constraint
starting from period t forward and using R (st ) = 1 we ...nd
X X Q (sj jst )
1
t
B s = Pj s j c s j W sj 1 ` sj 1 n
sj (A3)
j=t
R (sj )
sj jst
and solving the economy-wide budget constraint starting from period t forward,
XX
1
m (sj )
t j j
B s = Px s A (sj ) 1 ` s Q sj jst (A4)
j=t sj jst
(1 ) d (st )
Using (5), (11) and (14) into (18) we ...nd that the tax rate on foreign bonds is uniquely
determined by
1
t+1 Uc (st+1 ) R (st ) d (st+1 ) Z (st+1 )
1+ s = : (A6)
Uc (st ) R (st+1 ) d (st ) Z (st )
There are three cases to consider: 0 < < 1; = 0, and = 1. These are, respectively,
a partially sticky economy, a exible prices economy, and a fully sticky prices economy. All
three cases have R (st ) = 1; d (st ) = 1, (st+1 ) determined from (A6) and holdings of
foreign bonds B pinned down from (A4).
Case I. A partially sticky economy, i.e. 0 < < 1. In this case, the policy that
decentralizes e is uniquely determined. Since there are sticky ...rms, the price level at period
a
s
t depends on st 1 . Hence, Pt (st 1 ) = Pt (i; st 1 ) for all i. At period 0, P0 (s 1 ) = P0 is given.
Then, (A1) can be expressed as
t Pt (st 1 ) X
Pt+1 s = t)
Uc st+1 st+1 jst (A5)
Uc (s t+1 t
s js
29
s
Given e and P0 (s 1 ) = P0 ; (A5) determines recursively the price level Pt (st 1 ) that justi...es
a
the Friedman rule R (st ) = 1. Given the sequence of prices, (A2) determines the required
money supply, and the equilibrium prices Q (st+1 jst ) follow from (5).
The pricing equation (14) together with the condition Pt (i; st ) = Pt (st 1 ) can be used
to pin down the foreign-input price Pm (st ) : Given Pm (st ) ; (11) determines the nominal
exchange rate E (st ) and the home input price Px (st ) : The wage rate W (st ) follows from
(10), and (4) pins down the labor tax rate n (st ). Finally, the stock of domestic debt B (st )
is obtained from (A3).
By construction, exible and sticky ...rms ...nd it optimal to set a common price Pt (st 1 ) :
Since all the conditions of an equilibrium are satis...ed, the relaxed Ramsey allocation is
implementable with a unique policy ! R : Thus, e is also the Ramsey allocation.
a
Case II. Flexible economy, i.e. = 0: In this case, as in Lucas and Stokey (1983),
monetary policy fE; M g and domestic bonds policy fBg are indeterminate. Given the
allocation e; the real wage W (st ) =Pt (st ) is determined from (10) and (14). Then, (4) pins
a
down the labor tax rate n . To see the indeterminacy of monetary policy, note that by
symmetry, all ...nal goods ...rms set a common price Pt (st ) given by (14). This price, in
general, depends on the current shock st . Speci...cally, given the allocation, any sequence
of prices satisfying (A1) can be an equilibrium path for the price level. In particular, the
equilibrium price sequence Pt (st 1 ) obtained in case I can be implemented as an equilibrium
in a exible prices economy. Then, (14) and (11) determines the nominal exchange rate
t
E (s ), and given the prices and allocation, (A3) pins down the bonds holdings B: The rest
of the equilibrium prices are obtained as in case I.
Case III. Fully sticky prices economy, i.e. = 1. This case also features an indeterminacy,
but of a very di¤erent nature. Here, all policy instruments except labor taxes and nominal
s
exchange rates are uniquely determined. First, given P0 and the allocation e, the sequence of
a
t 1 t
prices Pt (s ) is obtained from (A5). The money supply M (s ) and the equilibrium prices
Q (st+1 jst ) are given by (A2) and (5). To see the indeterminacy, note, ...rst, that in period 0
any combination of n and W0 satisfying (4) can be implemented as an equilibrium. Second,
0
using R (st ) = 1; (4), (5), (10) and (11) into (15) for all t 1, and noting that Pt (st 1 ) is
independent of st , we obtain
X U` (st ) =Uc (st ) d (st ) 1
t
1= s (A6)
1 st jst 1
(1 n (st ))
Z (st ) A (st ) (1 )1
where
Uc (st ) [c (st ) + g (st )] (st jst 1 )
st = P t
:
st jst 1 Uc (s ) [c (st ) + g (st )] (st jst 1 )
Any labor tax rate n (st ) satisfying (A6) can be decentralized. In particular, any pair of
labor tax policies n (st ) and n (st ) satisfying (A6) decentralizes the Ramsey allocation.
1 2
According to (5), however, they induce di¤erent nominal wages W1 (st ) and W2 (st ), which,
by virtue of (10) and (11) induce di¤erent nominal prices fE1 (st ) ; Px1 (st ) ; Pm1 (st )g and
fE2 (st ) ; Px2 (st ) ; Pm2 (st )g.
30
In all three cases I, II and III, there are prices and policy instruments ! R such that
relaxed Ramsey allocation e is an equilibrium allocation. Therefore, e is indeed the Ramsey
a a
R
allocation a :
31
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