Download Fiscal Policy in a Currency Union at the Zero Lower Bound ∗

Fiscal Policy in a Currency Union at the Zero Lower
Bound ∗
David Cook†and Michael B Devereux‡
July 2015
Abstract
A combination of sticky prices and home bias imply that asymmetric demand shocks
can lead to disparate outcomes for inflation and the output gap. In a currency union,
monetary policy cannot be used to address these disparities generating a role for fiscal
policy. This paper examines the use of fiscal expenditure to address regional disparities
with a focus on the experience of an economy at a liquidity trap.
Keywords:Liquidity Trap, Monetary Policy, Fiscal Policy, International Spillovers
JEL: E2, E5, E6
∗
The authors thank the Hong Kong Research Grants Council for support under CERG GRF 690513.
Devereux thanks SSHRC, the Bank of Canada, and the Royal Bank of Canada for financial support as well
as support from ESRC award ES/1024174/1.
†
Hong Kong University of Science and Technology, [email protected]
‡
CEPR, NBER, University of British Columbia, [email protected]
1
Introduction
Over the last decade, developed economies have been operating with very low interest rates
which has constrained monetary policy. Constraints on the use of monetary stabilization
raises the possibility of alternative stabilization instruments including fiscal policy. Optimal
fiscal stabilization may be complicated in a currency union subject to asymmetric shocks.
First, when aggregate demand conditions are sufficiently severe to constrain monetary policy,
fiscal policy can be used to stimulate inefficiently low demand. In addition, if there are
asymmetries in demand fiscal policy may be targeted toward locations with low demand.
Given the absence of exchange rate adjustment to allocate demand efficiently across different
regions, fiscal policy might also be called upon to help.
This paper considers the implementation of fiscal policy in a currency union modeled
as a set of regions which face asymmetric shocks with a single monetary policy. This paper
will concentrate on examing the role of demand shocks in a New Keynesian model when
countries have a bias toward home goods. Its understood (see Woodford) that aggregate
fiscal policy can play a role in off-setting the impact of negative demand shocks that force
the economy to the zero lower bound. However, in a currency union subject to asymmetric
shocks, regional fiscal policy should also be used to address regional disparities. This paper
examines the degree to which these fiscal goals may conflict in implementing policy. Consider
the post GFC European economy with more extreme downturns observed in some peripheral
countries than in the core countries of the Eurozone such as Germany. We ask whether it is
optimal to apply expansionary fiscal policy in both regions or whether it would be preferable
to concentrate fiscal demand in the more badly affected regions.
We find a key factor affecting the answer to this question is the degree of home bias
in consumption. If domestic demand is concentrated on domestic goods, then asymmetric
shocks will result in greater regional business cycle disparities leading to a greater need to
concentrate spending in a particular region. Moreover, the spillovers of fiscal policy from
one region to another are reduced as home bias concentrates any of the effects of fiscal policy.
Thus, fiscal policy will be less effective in addressing demand shocks in another region. This
paper examines the relationship between fiscal multipliers and home bias to demonstrate
this.
Conversely, if the economy is particularly vulnerable to the aggregate downturn, then
meeting aggregate optimal spending goals may require expansion across the union. For
example, if demand is sufficiently interest sensitive, then a relatively large fiscal expansion
2
may be required. This will tend to encourage fiscal policy expansion in both regions. If
shocks are persistent, the effects of demand shocks will be intense which will also increase
the likelihood of possible spillovers.
We identify these results by characterizing social welfare as a second order approximation
to the utility of the residents of the currency union. We focus on demand shocks which can
move inflation and the output gap in the same direction. Thus, there is no direct trade-off
between stabilizing the two given monetary policy. However, fiscal policy cannot an efficient
equilibrium as fiscal spending changes the mix of goods. Likewise, asymmetric shocks distort
the efficient allocations pushing output gap and inflation differentials in the same direction.
Fiscal policy that directs spending toward low demand region can offset both. However, this
also will creates additional distortions, so perfect allocation cannot be achieved.
As noted by, Corsetti, Keester and Muller (2012) and Cook and Devereux (2015), a
currency union does contain a commitment to maintaining price level differentials. Thus,
the optimal rules are dynamic in nature, even when no fiscal commitment is available.
However, we do find a simple rule that directs fiscal demand toward regions with severe
output gaps will closely match optimal reallocative policy.
A number of papers in the New Keynesian tradition have examined fiscal policy issues
when monetary policy is constrained. Christiano, Eichenbaum and Rebelo (2011) examine
the determinants of fiscal multipliers at the zero lower bound. Woodford (2010) examines
the optimal fiscal multiplier. Fujiwara and Ueda (2013) identify fiscal multipliers in an open
economy with flexible exchange rates; Cook and Devereux (2010, 2011a) examine optimal
fiscal policy. Farhi and Werning (2012) examine fiscal multipliers in a currency union; in and
out of the zero lower bound. Hettig and Mueller (2015) examine the coordination of fiscal
policy in a union of many small economies at the zero lower bound. Blanchard, Erceg and
Linde (2015) examine the welfare gains from expansion in a two country sticky price model.
In some sense, we build on this by characterizing optimal policy.
2
A two country model
Consider a currency area made up of two regions. We assume that both of these regions
have permanently committed to a single currency unit. In each country, households consume
both private and government goods, and supply labor. Denote the countries as ‘home’
and ‘foreign’, with foreign variables denoted with an asterisk superscript. The population
of each country is normalized to unity. Monopolistically competitive firms in each region
3
produce differentiated goods with constant returns to scale technology. Regional governments
produce government goods which are distributed uniformly across households withing the
region. Governments have access to lump sum taxation. Complete asset markets allow full
insurance of consumption risk across countries. There is an implicit risk free interest rate
which is common across the currency union. Firm’s production and supply is constrained
by sticky prices.
2.1
Households
Utility of a representative infinitely lived home household evaluated from date 0 is:
0 = 0
∞
X
=0
  ((    ) −  ( ) + ( ))
(1)
where felicity is the functions  ,  , and  represent the utility of the composite home
consumption bundle (  ) disutility of labor,  ( ) and utility of the government good
( ) respectively with   1. The variable   represents a demand shock to preferences
(12  0).
The composite consumption consists of a geometric average of home and foreign goods.
2
1−2
 = Φ  
where Φ =
¡  ¢ 2
2
(1 −
 ≥1
¡ ¢ 
) 2   is home consumption of the home country produced com2
posite good , and  is home consumption of the foreign produced composite good. We
assume   1 indicating realistic home bias for domestic goods.
Consumption aggregates,  and  are composites, defined over a range of home and
foreign differentiated goods, with elasticity of substitution   1 between goods.
1
⎡ 1
⎤ 1−
Z
1
 = ⎣  () 1− ⎦ 
1
⎡ 1
⎤ 1−
Z
1
 = ⎣  () 1− ⎦ 
0
0
The demand for good  in region  =   is
 ()
=

µ
 ()

¶−
where the price indices for home and foreign goods are:
4
1
⎡ 1
⎤ 1−
Z
 = ⎣  ()1− ⎦ 
1
⎡ 1
⎤ 1−
Z
 = ⎣  ()1− ⎦ 
0
0
2
1−2
while the aggregate CPI price index for the home country is  =  
2
1−2
foreign country is ∗ =  
and for the
. In each region, government spending has complete home
bias; agents only get utility from spending on the domestic good. Government demand for
each variety of home goods has price elasticity , the same as that for private spending.
The household’s implicit labor supply at nominal wage  is:
 (    ) =   0 ( )
(2)
Optimal risk sharing implies
 (    ) =  (∗   ∗ )
where  =


is:
2.2
∗
=  (∗   ∗ )−1 

(3)
the terms of trade. Nominal bonds pay interest,  . Then the Euler equation
¸
¸
∙
∙ ∗
∗
  ∗+1 )
1
  (+1   +1 )
  (+1
=  

=  
∗

+1  (    )
+1
 (∗   ∗ )
(4)
Firms
Each firm  employs labor to produce a differentiated good.
 () =  ()
Profits are Π () =  () () − −1
  () including a subsidy to labor to eliminate steady

state first order inefficiencies.Price setting follows Calvo with probability of price adjustment,
1 − . Reset prices are e () :
e () =

P


=0
P
+
+ () +
+ ()
=0
where the stochastic discount factor + =
+ () + ()
 (+  + )

.
 (   )
+

(5)
The aggregate home price
index follows:
1
1−
1− 1−
 = [(1 − )e
+ −1
] 
5
(6)
and foreign price analagously.
2.3
Market Clearing
Equilibrium in the market for good  as
 () =
µ
 ()

¶− ∙
¸
 
 ∗ ∗
 + (1 − )
 +  
2 
2  
where  represents total home government spending. Aggregate market clearing in the
home good is:
 =
Here  = −1
R1 ³  () ´−

0
R1
 
 ∗
 + (1 − )  ∗ +  
2 
2 
(7)
 () is aggregate home country output, where we have defined  =
0
 It follows that home country employment (employment for the representative
home household) is given by  =
R1
() =   
0
The aggregate market clearing condition for the foreign good is
 ∗ ∗
 
 + (1 − ) ∗ ∗ + ∗ 
∗
2  
2  
R1
R1 ³ ∗  () ´−
where ∗ = ∗ () =   ∗  and ∗ =

∗
(8)
  =
0

0
An equilibrium in the world economy with positive nominal interest rates may be de-
scribed by the equations (3), and then (2), (4), (5) and (6) for the home and foreign economy,
as well as (7) and (8). For given values of  and ∗ , given monetary rules (to be discussed
below) and given government spending policies, these equations determine an equilibrium
sequence for the variables  ∗    ∗       ∗  e  e∗    ∗  and   ∗ 

3
3.1

The Economy
Flexible Price Economy

Define  ≡ − 
as the inverse of the elasticity of intertemporal substitution in consump
00
00
tion,  ≡ −  
as the elasticity of the marginal disutility of hours worked and   ≡ −  0
0
as the elasticity of marginal utility of public goods. In addition, we assume that   =   1,
consistent with empirical evidence (see Yogo, 2004) Finally,  =
of a positive demand shock in the home country. Define  =
6




ln(  ) is the measure
is the steady state share of
consumption in output.
The shocks in the model are assumed to follow a Markov process. We assume preference
shocks are unanticipated and reverts back to zero with probability 1 −  in each period.
Wicksellian, or ‘natural’ interest rates, defined as the equilibrium interest rates from a
flexible price equilibrium of the world economy, where there are no monopolistic distortions
with optimal fiscal policies with access to lump-sum taxes. This rule is characterized by the
optimal trade off:
 0 ( ) =  0 ( )
(9)
Note that natural interest rates are equivalently defined as the value of the nominal rate less
expected PPI inflation, (i.e.  −  +1 for the home economy) in a flexible price economy.
That is, the PPI based real interest rate that would hold with flexible prices.
For any variable  , define the world average and world relative level, 
 =


=
 −∗
.
2
 +∗
2
and
Define  as the percentage deviation of the flexible price equilibrium from
the non-stochastic steady state. Define    ≡ ((2 − ) + (1 − )2 )  1, and ∆ =
 + (1 −  ) +   1. In addition, define  ≡
(−1)
.

The parameter, 0 ≤  ≤ 1,
measures the intensity of home bias. In the absence of home bias,  = 0; under full home
bias  = 1.We can solve as in Cook and Devereux (2011a)

 =
where ∆ ≡
∆
.

 
 
 =

∆
∆ 
Note that, approximated around the steady state, up to a first order, ≈
 , ∗ ≈ ∗  so the labor gap for each country will stand in for the output gap From (9) we
solve:
 
  


 = −  = −

 ∆ 
If the preference shocks,  and ∗ , following first order process with persistence, , then
natural interest rates of the home and foreign economy in a competitive equilibrium with
optimal government spending in both countries as in (9) are defined as:
 =  +
µ
¶
 
 
 (1 − )
 +
+ 
∆ 
(10)
∗
µ
¶
 
 
 (1 − )
 −
+ 
∆ 
(11)
=+
where  is the rate of time preference/non-stochastic steady state. Since capital markets
are integrated, shocks to natural interest rates move together across countries unless the two
7
economies are completely closed to trade (i.e.  = 2). This implies

(1 − ) 

+
 
= (1 − )

∆ 

= +



We can see that risk sharing insures that in the absence of home bias (i.e.  = 0) there are
no natural interest rate differentials.
3.2
Dynamic Model
A sticky price log-linear approximation of the model in terms of inflation and output gaps
(as in Clarida et al., 2002, and Engel, 2011) We linearize around an efficient, flexible price
allocation with zero inflation and there is an optimal subsidy which offsets monopoly distortions, 1 . Let 
e be the percentage deviation of a given variable  from the efficient flexible
price equilibrium. Thus, 
e is interpreted as a ‘gap’ variable.
3.2.1
Aggregate Economy
Despite the presence of multiple regions with home bias, the world average of the economy
is described by the canonical New Keynesian Phillips Curve and Euler Equation.
© 
ª

e − z · 
e
+   

 = ( + ) 
+1
¡
¢


 (e

e
e
e
+1 − 
 ) =   −   −  +1
+1 − 
 ) −  (
where  indicates price stickiness  =
for government spending  ≡


(1−)(1−) 2
,

(12)
(13)
where intertemporal elasticity adjusted
, and the zero inflation fiscal multiplier is z =

.
(+)
These equations do not depend on the home bias parameter . Simplify notation by defining

e
 
 ≡ (1 −  )e
Monetary policy is given by the rule
 = Max(0  +   )
(14)
This set of equations has no endogenous dynamics. In equilibrium, all of the endogenous
1
Assuming the possibility of future constraints on monetary policy as a zero-probability event, see Adam
and Billi (2006, 2007), and Nakov (2008).
2
Throughout we assume   (1−)(1−)
z which is necessary for stability under passive monetary policy.

This puts a lower bound on the degree of price stability.
8
variables are functions of the exgonous variables including the aggregated demand shock 

and the fiscal gap, 
e
 .
3.2.2
Regional Disparities
Likewise, we can write the separate dynamics of the world ‘relative’ position variables in the
form of the canonical New Keynesian economy.:
© 
ª

e − z 
e
+  

 = ( + ) 
+1
(15)
¡ 
¢


e
e
e
  (e
+1 − 
 ) −   (
+1 − 
 ) =  −  −  +1
Define ielasticity adjusted for expenditure switching as    ≡


(16)
 1. Relative demand
can be more sensitive to changes in the intertemporal price through the response of the
terms of trade and expenditure switching to intertemporal price differentials. Define 
e
 ≡
(1 −  )e
 .
Notice this system does not include a term for monetary policy. The currency union
insures that the same interest rate applies region wide.Since there is only one nominal interest
rate, the relative interest rate equations for nominal bond rates do not impose any additional
constraints on the model. Unfortunately, these equations have multiple solutions as shown
by Benigno, Benigno and Ghironi (2007) . However, the solution is pinned down by the
additonal condition on the dynamics of the real exchange rate:
  −  −1 = − 

(17)
Due to sticky prices, adjustments in the terms of trade only occur through inflation differentials (see Benigno, 2004) shows that a fixed exchange rate imposes another initial condition
on the dynamics of the terms of trade in a single currency area.
Equilibrium in goods and financial markets imply
£ 
¤
£ 
¤

 
1 
=  
e
 =  
e

e − 
e − 
 −
 −
2
∆
1− 
(18)
But we can combine (??) (rewritten in ‘gap’ terms) with (17) to obtain a separate relationship between inflation and the output gap implied by the single currency area. Note,
that in the presence of home bias (  0 )
9
This is:


¶
µ
¶
£ 
¤
£ 
¤
 
 


+ 2  
= −2  
e −

e −1 −

e − 
e−1 − 
∆ 
∆ −1
¶
µ
£ 
¤
1


∆
e −
 − ∆
= −2  ∆e
1− 
µ
(19)
(20)
This replaces (16) as an equilibrium condition linking relative demand, represented by the
effect of output gaps on the terms of trade, to relative inflation. Equation (19) is dynamic;
relative inflation and output gaps will be endogenously persistent. In a single currency
area, the terms of trade change only through slow changes in domestic prices. Thus, the
aggregate currency area and the relative regional positions have fundamentally different
dynamics. Asymmetric shocks will have persistent distributional effects after the aggregate
effects dissipate.
Suppose that government fiscal gaps follow the same Markov process as demand shocks.
The dynamics of the terms of trade will follow the form. ith a single currency area, this
causes a terms of trade depreciation. The stable solution for the terms of trade is given as:
∙



  =  −1 −  ·
·  · 
e
+

 + 
1−
where
0 ≡ n

z

2z 
o=½
Ξ
+ (1 + (1 − )) − 
+
2
defining the parameter Ξ =

2z 
0 =
=

2
¸
2Ξ
1−
2
+ (1 − ) +
(21)
√
[Ξ+{1+}]2 −4
2
¾ 2
¢
¡ 
  + 1 . The root
q
[Ξ + {1 + }] − [Ξ + {1 + }]2 − 4
2
1
. It is straightforward to show that the root is real, positive and within the unit interval.
Persistence, (Ξ), is negatively related to Ξ.
0 (Ξ) =
1
(Ξ + {1 + })
[1 − q
]0
2
2
[Ξ + {1 + }] − 4
Intuitively, the larger is the impact of a shock on inflation, the less persistent will be changes
in the terms of trade. When prices are less sticky,  will be larger and Ξ will thus be larger
10
and shocks will be more persistent. Prices and terms of trade will adjust more quickly. When
 is larger, labor supply is less elastic and the output gap has a bigger impact on inflaton.
Most interestingly from the perspective of this paper, the parameter Ξ is negatively
associated with home bias. The closer  is to 2, the smaller will be  and Ξ and the greater
the persistence in the shock. When a given appreciaton in the terms of trade will translate
into less expenditure switching that will have less impact on inflation
Note that in the unit intertemporal elasticity case,  = 1,  is invariant to the degree of
home bias. Thus, the system of equations (15), (16) and (17) are affected by only through

 .
4
Fiscal Multipliers
Examine the marginal effect of a fiscal gap in the home economy. Suppose that 
e  = 2
and 
e ∗ = 0, so that 
e
e
 = 
 =  Assume that government spending follows a Markov
process so that 
e
+1 =  with probability  and 0 otherwise. We can identify the marginal
impact of government spending on the aggregate economy under to separate regimes. In the
first, the interest rate is set according to the Taylor rule.
(1 − )(1 − ) + ( − )
z
e
 + 
(1 − )(1 − )z + ( − )
(1 − )(1 − ) + ( − )

z  
z1
=
(1 − )(1 − )z + ( − )


e
= 

(1−)(1−)
z

− Due to price stickiness, the multiplier under the Taylor rule is larger than
the zero-inflation multiplier z. However, the inflationary effects of aggregate government
spending increases the interest rate gap whicdh crowds out private spending.
We can also examine the multipler if the authorities are facing negtive demand shocks
which push the natural rate below the zero lower bound.


e
= 
z
e

 + 
(1 − )(1 − ) − 

z  1  
z
=
(1 − )(1 − )z − 
In the absence of an interest rate response the multiplier is larger than 1.
Fiscal policy in one country will also create differential output responses across the currency area. Since regional disparities are invariant to monetary policy, so the fiscal multiplier
of relative system
e



e

is invariant to the zero lower bound. However, the endogenous dy11
namics of regional disparities also create a dynamic multiplier. Combine (17) with (21)
µ
e
 


−  −  
e

1−
¶
=
µ
which implies

e

 
e
−1
¶
∙
¸




−1



−
e −1 − ·
·  · 
e +
−  
1−
2  + 
1−
∙
¸








−1


=
+
·  · 
e +
+ 
e −
− 
e −1 −
·
 (1 − )
 (1 − )
 2  + 
1−
¸
¸
∙
∙


−1




+ [1 − ·
·
e
] · 
e
=  
e
−1 − 
−1 −
 + 1−
 (1 − )
2  + 
2
 (1 − )
e

−1
Proposition 1 The impact multiplier   ≡ [1 −

2
·

]
+
is between 0     1.
Proposition 2 When  = 1, the impact fiscal multiplier is invariant to the degree of home
bias. When   1, the impact multiplier differential,   , will be larger when home bias
is more intense.
Suppose that 
e
e
 =  and 
+ =  with prob  or zero for all  ≥ 0. Conditional
on the government spending continuing we can write

e
+
!
#
à 
X



 + 
·
= 1−

2  + 
=0
"
We call the multiplier during the duration of the spending the expanison phase multiplier
 ()
h
³P
´

 
Proposition 3 The expanison phase multiplier  () = 1 −

·
=0
2
declining in  and between 0   ()   ( − 1)  1.

+
i
is
The impact of government spending differentials on output dissipates over time. The
region with a more concentrated level of government spending will experience a slow terms
of trade appreciation. This will shift expenditure to the other region. However, the impact
on local output always exceeds that on domestic output.
The post expansionary multiplier depends on how long the expansion lasts. Consider if
the expansion ends in period  +  + 1. Then, we can write output as:
£ 
¤


e
e+ − 
e
() − 1]  0
+ = [
++1 =  


() − 1]  0

e
++ =  [
12
So, the post expansionary multiplier on output differentials is negative. After the expansion,
the expanding economy will have an overvalued terms of trade. Demand for their products
will be reduced.
If the economy is at the zero lower bound when the expansion occurs, fiscal expansion
in one country will increase output in the other country. Define the cross-region multiplier,
e


=  −  

Proposition 4 The cross region multiplier is always positive at the zero lower bound, 
()−
 ()  0
If the expansion occurs outside the zero lower bound, the cross-region multiplier in a
currency union may be more complicated. A fiscal expansion concentrated in one region
will lead to a real appreciation increasing demand for the other regions goods. However, the
fiscal expansion will increase aggregate inflation which will impact aggregate interest rates
for all regions of the currency area. The net effect of these counter-veiling effects on crossregional demand are ambiguous. The previous proposition shows that if monetary policy
is sufficiently passive (i.e. zero interest response) then the positive spillovers will dominate.
However, if aggregate monetary policy is sufficiently active and home bias weakens the
expenditure switching effects of exchange rate appreciation, then the negative spillovers will
dominate.
Proposition 5 If the monetary policy rule implements a zero inflation equilibrium, there
exists an  such that the cross-region multiplier during the expansion is negative.
Clearly, in the periods after the expansion, the cross region multiplier will be positive as
 ( + ) will be negative for any  ≥ 0.
5
Relative Demand Shocks
Regional differences in the natural interest rate generate inflation and output dispersion by
concentrating demand within a particular region. Abstract from fiscal policies. Consider if
government spending gaps areis zero. Output follows the process
¸ ∙
¸



−1

+ 1−
·
 (1 − )
2
 (1 − )
1 −  + (1 − )

ª 1
0  1 − = ©1
2
Ξ + 1 −  + (1 − )
2

e

∙
=  
e
−1 −
13
A natural rate shock following a Markov process that dissipates (ex post) at period  +  + 1
has an impact

e
+
! #
à 
X



·
= 1−

2

 (1 − )
=0
"
where the inequality follows from the same logic as Proposition 3. In period  +  + 1

e
++1

e
++
à  !
¸
X



+

= [−
]
0

 (1 − )
2

(1
−
)

=0
à  !

X


=  [−

]
2  (1 − )
=0
∙
=  
e
+ −
à  !
X

− [

]  0
2
=0
So a positive demand shock will have positive but diminishing effects on output until the
shock dissipates, then output effects reverse.
Inflation follows


=

−1
∙
 − 
−1
+ · 
(1 − )
¸
Again, a natural interest rate differentials shock increases inflation differentials during the
period of the shock, then reverses after the shock dissipates.
5.1
Countercyclical Policy
Consider the impact of policy rules. The government authorites implements endogenous
spending differentials that take into account economic decisions.

e
e
 = −Φ · 

←
→
Φ
In the case of endogenous fiscal policy, we can redefine the parameter Ξ = Ξ · 1+z
 Ξ..
1+Φ
The stable solution for the terms of trade has the same form as (21):
←
→
←
→
  =   −1 −  ·
14


(1 − )
(22)
where the initial impact of demand shocks on the terms of trade and its persistence are a
←
→
function of Ξ
h←
i rh←
i2
→
→
Ξ + {1 + } −
Ξ + {1 + } − 4
←
→ ³←
→´
1
0   Ξ =
2
←
→
←
→ ³←
→´
2Ξ
⎫ 2
 Ξ
= ⎧
r
2
←
→
⎨ ←
⎬
Ξ
+{1+}
−4
[
]
1−
1 →
Ξ + 2 + (1 − ) +
2
⎩2
⎭
←
→ ←
→
where as shown previously  0 ( Ξ )  0 and as implied by the proof to proposition ( 2). Note
←
→
Ξ is decreasing in Φ. So, the more counter-cyclical is fiscal policy (i.e. the larger is Φ) the
smaller will be the initial impact of demand shocks on the terms of trade as re-allocation of
public sector demand across regions will offset changes in private sector demand. However,
demand shocks will have less impact on inflaiton differentials, so inflation will adjust more
slowly.
We can write the dynamics of output as
µ
 
e

¶ ←
→



−1


−  
·
−
e −1 −
(1 − )
2 (1 − )
¶ "
µ
←
→#

¢
¡

←
→


−1


e
(1
+
Φ)

e
−
+
1
−
·
=


 (1 + Φ) 


−1
(1 − )
2
(1 − )
h
←
→i

∙
¸

1
−
−1
2
←
→ 



e−1 −
·
+

e =  
 (1 + Φ) (1 − )
(1 + Φ)  (1 − )


−  
−
e

(1 − )
¶
←
→
= 
µ
 
e
−1
Proposition 6 Counter-cyclical fiscal policy differentials will reduce the immediate impact
of demand shocks on the output gap.
Inflation follows


∙
 ¸
←
→  ←
→ 
 −  −1
=   + 
(1 − )
Again, a natural interest rate differentials shock increases inflation differentials during the
period of the shock, then reverses after the shock dissipates.
15
6
Optimal Fiscal Policy
As shown in Cook and Devereux (2011a), a second order approximation to an equally
weighted world social welfare can also be constructed in world averages and world differences. Welfare for any period  is written as:

2
2 
2 
− (e

− (
e
− (
e
− (e

e
 )
 )(
 ) ·
 ) ·
 )
2
2
2
2
  2

(  ) − ( 
e
)2
−(e

 )−
 )(
2
2 
2
 = −(e

 ) ·
(23)
where       are defined in the Appendix. Thus, the social welfare function faced by
the policy maker depends upon output gaps, inflation rates, fiscal gaps, and the interaction
between these variables.
Given this, cooperative optimal policy maximizes the objective function

"∞
X
  +
=0
#
£ 
¤


e
  − ( + )e
 =  +  
 −   +1

 +  · 
£ 
¤


e
+ 
 −   +1
   − ( +  )e
 +  
£
¡
¢¤

e
e
e
e − 
 (e
+
+1 − 
 ) −   − 
+1 − 
 ) −  (
+1

∙
¶
µ
¶¸
µ
£ 
¤
£ 
¤


−1





+   + 2  
− 2  
e −
e −1 −
e − 
e−1 − 
(1 − )
(1 − )
+  
where the constraints are the equation describing the dynamics of the economy and a final
condition that  ≥ 0
6.1
Optimal Policy Under Discretion
We first examine optimal policy under discretion. The system of equations that characterize the optimal differentials 
e
 can be examined separately from those governing optimal
aggregate policy, 
e
 .
6.1.1
Aggregate Economy
The first order conditions describing optimal policy for the aggregate economy are
16


−e

e
 − 
 = ( + )  +  


 
e

 + e
 =   +  


 =  

 = 
The constraint on  at the zero lower bound implies that either the shadow value,   , is
zero or the zero lower bound binds,  .
When   is zero and policy rates are unconstrained by the zero lower bound,  = e
 price stability is the cooperative optimal policy under both commitment and discretion,
(see Benigno and Benigno, 2003); the aggregate fiscal gap will be zero along with output gap
and inflation. Intuitively, if policy is determined relative to an initial steady state without
monopoly distortions, and there are no mark-up shocks, optimal cooperative policy will
close all gaps, whether under discretion or commitment. A policy of price stability can be
implemented by setting nominal interest rates equal to natural interest rates as defined in
(10) and (11). Note that in order to implement this policy, it is necessary that there be
an interest rate feedback rule on inflation or other endogenous variables, in order to avoid
indeterminacy (see e.g. Gali, 2008, and Benigno and Benigno, 2008). For related discussion
on determinacy issues in open economies, see
When the zero bound on interest policy binds, the optimal government spending is

(1 − ) 

+
[(1 − ) + ( + )] (1 − )
=


[(1 − ) ( +  ·  )  + ( + {
− 1} )] [(1 − )(1 − ) − ( + )]
 = − · e = − ·


 1, so its clear that optimal fiscal gaps are
In a liquidity trap, the fiscal multiplier 
negatively associated with demand shocks.
Figure illustrate the response of the aggregate demand shock to a Markov demand shock
that reduces the natural rate by 800 basis points, We identify the stable solution numerically.
The benchmark parameterization of preference parameters include subjective discount factor
is  = 099; the inverse of the Frisch elasticity of labor suppy,  = 1; the inverse of the
intertemporal elasticity of substitution,  = 2; and the elasticity of substituton between
differentiated goods,  = 5 The steady-state government share of output,  = 2, and the
persistence of the demand shock,  = 075; and price stickiness,  = 0825. The benchmark
measure of home bias,  = 15 is associated with steady state imports consisting of 20% of
17
GDP.
Figure (1) illustrates the response of regional differentials to a demand shock in the
home region. We examine the response to a Markov shock that reduces the average natural
interest rate by 200 basis points, e =  − 02. The natural interest rate reverts to  with
probability  or continues at  −02 with probability 1−. The benchmark parameterization
of preference parameters include subjective discount factor is  = 099 and the inverse of the
Frisch elasticity of labor suppy,  = 1; and the elasticity of substituton between differentiated
goods,  = 5 The steady-state government share of output,  = 2, aand price stickiness,
 = 0825; and shock persistance,  = 075). Figure (1) shows the response of the economy
under different degrees of intertemporal elasticity of substitution: low elasticity,  = 4;
medium elasticity,  = 2; and low elasticity,  = 1.
. In the scenario described, the shock
lasts (ex post) for 8 periods before reverting to steady state. The size of the shock drops 800
annualized basis points below the steady state value. Panel B reports the relatiohsip between
the degree of demand elasticity and the size of the multiplier. Fiscal policy will moderate the
negative effect of the demand shock on inflation. Persistent fiscal policy response will also
moderate expected disinflation. This will reduce real interest rates stimulating demand. The
response of the demand depends on the degree of intertemporal elasticity. The aggregate

fiscal multiplier, 
is close to 1 when elasticity is low; about 1.5 under medium elasiticity;
and close to 3 when intertemporal elasticity is high. Panel C shows the counter-cyclical
response of optimal aggregate fiscal policy. Given greater, optimal fiscal policy will be
larger for two reasons. First, the more persistent shock leads to a stronger business cycle
contraction in any period. At the zero lower bound, a persistent shock is associated with
expected disinflation (see Panel E) which will raise real interest rates and reduce demand
and lead to a more negative output gap (Panel D). The role for counter-cyclical policy is
larger. Second, the effectiveness of fiscal policy is larger as measured by the multiplier.
6.1.2
Regional Differentials
The first order conditions governing regional differentials are:



−e

e
 ) =   ( +  ) − 2 [  −  +1 )
 − (



 
e

 + (e
 ) =    − 2 [  −  +1 )
(25)


 +   =  
(26)
18
(24)
The first order conditions can be solved forward:
"∞
"∞
#
#
X
X
¡
¢
¡
¢
+

)
(2z




 ∗∗   
e
b
 ∗ ( +  ) 

b
=
 − 
 −  
 + (e
 ) +  


(2z
+
)

=0
=0
where  ∗ ≡
1

1+Ξ
where  ∗∗ =
1
.
1+z  Ξ
The optimal choice of fiscal policy implies a trade-
off between the various distortions. If relative demand shocks are creating distortions in the
relative output gap and relative inflation, then optimal government spending gaps cannot be
zero. Since autoregressive relative demand shock shift the output gap and inflation in the
same directions, the fiscal gap should move in a counter-cyclical direction.
We search for a solution to the set of equations
© 
ª
− {  −  −1 }  
= ( +  ) 
e − z 
e
+   

+1

© 
ª
− {  −  −1 } = ( +  ) 
e − z 
e
−  [ +1 −   ]

where optimal fiscal policy accords with ( 24), (25), and (26). We search for solutions for
the dynamics of the real exchange rate.
  =  −1 −  ·


(1 − )
We identify the stable solution numerically. The benchmark parameterization of preference
parameters include subjective discount factor is  = 099; the inverse of the Frisch elasticity of
labor suppy,  = 1; the inverse of the intertemporal elasticity of substitution,  = 2; and the
elasticity of substituton between differentiated goods,  = 5 The steady-state government
share of output,  = 2, and the persistence of the demand shock,  = 075; and price
stickiness,  = 0825. The benchmark measure of home bias,  = 15 is associated with
steady state imports consisting of 25% of GDP.
Figure (2) demonstrates the response of regional differentials to a demand shock in the
home region. We examine the response to a Markov shock that reduces the average natural
interest rate by 200 basis points. This shifts the rate from an annualized 4 percent steady
state level to a natural rate of -4%. This negative rate lasts for 8 periods (though this is
not known ex ante) before reverting to the steady state level permanently. . We show the
response to the differentials under three cases: 1) no fiscal policy, 
e
 = 0; 2) rule of thumb
counter-cyclical policy with Φ = 2; 3) optimal fiscal policy as characterized by ( 24), (25),
and (26).
Figure (2), Panel A shows that a asymmetric demand shock leads to a natural interest
19
rate differential, with the rate falling more in the more badly affected home region. Note
that though the average rate falls by 200 basis points, the fall in the interest differential is
smaller due to the spillovers of the demand shock across regions. Figure (2), Panel B shows
the natural rate differentials lead to a real depreciation in the affected region. The shock to
demand in one region leads to a persistent depreciation. However, with a single currency,
the exchange rate adjustment can only occur through internal devaluation. Slowly, the real
depereciation intensifies until the 8th period when the trajectory reverses back toward steady
state. The natural interest rate differentials result in a change in optimal fiscal policy. Panel
C shows that optimal discretionary policy suggests the more affected country should have a
counter-cyclical fiscal response, expanding government spending to offset the impact of the
negative demand shock through the first 8 periods when the home country is experiencing
negative demand. Panel D and E show that this optimal response should not be so large as
to eliminate business cycle response. Both the output gap and inflation persistently decline
most sharply in the affected country through the period in which the demand shock has
an effect. After the shock reverses, the home country will continue to have a competitive
real exchange rate. Demand will switch back towards the home country which will enjoy
a temporary expansionand relatively higher inflation as the real exchange rate returns to
steady state. Consumption also declines in the affected countries.
Comparing across fiscal policy regimes it appears that the difference between optimal
discretionary fiscal policy and a naive rule of thumb counter-cyclical policy is not quantitatively large. Both will imply a basically static response to the demand shock in the different
regimes. By contrast, the absence of any fiscal policy response will result in a much larger
decline in the output gap differential. There is a relatively smaller response in regards to
consumption to fiscal policy. Apparently most of the benefit of fiscal policy may come from
smoothing labor effort.
The degree of home bias clearly affects the degree to which asymmetric shocks lead to
differential outcomes. If there were no home bias, the natural rate in each region would be
identical and there would be no regional differentials in any of the variables. We examine the
impact of a home centered demand shock on regional differentials under three parameterizations for home bias: 1) the low home bias case,  = 125, where imports make up about
37.5% of steady state consumption; 2) the benchmark case,  = 15, where imports make
up about 25%; and 3) the high home bias case,  = 125, where imports make up about
12.5%. The aggregate economy is hit by a shock that reduces the aggregate natural rate
by 800 annual basis points. If there were no home bias, this would be felt equally across
regions and regional differentials would be zero; in the complete home bias case,  = 2, the
20
demand shock would only affect the home economy and natural rate differentials would be
the full 800 points. In the intermediate cases we consider, natural rate differentials lie within
the range. Figure (3), Panel A shows that the response of natural interest rate differentials
which are as narrow as 200 basis points or as wide as more than 600 basis points in the very
closed economy case. Figure (3), Panel B shows that the degree of openness impacts the
equilibrium real exchange rate response with relatively closed economies seeing relatively
sharper responses over the short-term horizon. Given the larger interest differentials, the
impact of the shock on demand differentials wil be relatively larger. Panel F shows that the
inflation differential is also larger. Panel C shows that optimal response of the fiscal gap.
The greater the degree of home bias, the more concentrated demand shocks are on the home
economy and the more concentrated fiscal policy will be in the home economy. Interestingly,
this effect modifies the output differentials as well. Panel D shows that althugh the Benchmark output gap differentials are more severe in the Benchmark case than in the low home
bias case, the Benchmark and the high home bias case show similar output differentials.
First, there is a larger counter cyclical fiscal response to offset the larger demand shock,
but Proposition also shows that the fiscal multiplier will also increase with home bias. This
stronger counter-cyclical effect limits teh impact on the output gap. However, the fiscal
policy response means that the consumption gap will widen under strong home bias.
6.2
Optimal Policy Under Commitment
We also consider the choice of fiscal policy under commitment. Our main interest is fiscal
policy in a currency union at the zero lower bound. Commitment in this cases raises a
number of issues requiring non-linear solution methods which we will defer. However, unconstrained aggregate monetary policy will offset all demand shocks, so fiscal policy only
considers regional differentials. It may be useful to consider what policy under commitment
looks like under unconstrained monetary policy.
The first order conditions governing regional differentials under timeless commitment are:



−e

e
 ) =   ( +  ) − 2 [  −  +1 )
 − (



 
e

 ) =    − 2 [  −  +1 )
 + (e


(
 −  −1 ) +   =  
21
(27)
The solution for the endogenous variables will be a function of three state variables, the
current level of the exogenous demand shock, the lagged real exchange rate and the lagged
co-state variable, 
−1 , We solve this numerically under the Benchmark parameterization.
Figure (4) compares the response of regional differentials to a Markov natural rate shock
under commitment and discretion along with naive counter-cyclical policy. Optimal fiscal
policy under commitment shares a counter-cyclical path with optimal policy under discretion.
However, the dynamic path is different. The fiscal gap under commitment will expand and
then smoothly increase during periods of low demand.
Under committment, the fiscal
policy can internalize the impact of future fiscal policy on expected inflation. The planner
implements a plan to slow the disinflation by stimulating future demand. After the natural
rate reverts to steady state, the planner will allow the fiscal gap to converge back to zero. As
shown in Panel D, the accelerating path of the fiscal gap will cause the output gap to recover
more quickly. The expectations of a faster output gap recovery allows for less disinflation
(Panel E) while the terms of trade will adjust by less under commitment (Panel B). The
difference between commitment and discretion seems to matter relatively little for private
sector demand which is similar under both policies.
7
Optimal Regional Allocations
We can examine the different allocations to the home and foreign economy. For any home

∗


variable,  = 
 +  and any foreign variable is  =  −  .
Examing the allocations under optimal fiscal policy when monetary policy is uncon-
strained by the zero lower bound is straightforward, since for all variables 
 = 0 so the
home allocation will just be equal to 
 as reported in Figures (2) through (4) and the
foreign allocation would be the negative of that, a perfect mirror of the home respose.
In a liquidity trap, with constrained monetary policy, however, aggregate fiscal policy
will respond in a counter cyclical manner. The following Figure (5) shows the response of
the home economy to a negative home demand shock under different degrees of home bias.
As we might expect, there is a concentration of the decline in the natural interest rate in
the home economy under home bias. The home region experiences a decline in the natural
rate greater than 800 annualized basis points. Also, we see that counter-cyclical fiscal policy
will be concentrated in the home economy to the degree that the economy faces home bias.
The optimal home fiscal gap will increase to as large as near 4% under high home bias or
lower than 2% as under low ome bias. Fiscal policy will stabilize the home output gap and
22
inflation at similar levels regardless of home bias.
The following Figure ( 6) shows the response of the foreign economy under a shock to
the home economy. The decline in demand in the home economy spills over to the foreign
natural rate. The spillovers are larger when home bias is small. This will be associated
with more severe exchange rate appreciations in economies with more intense home bias.
Panel D and Panel E show that the spillovers of the demand shock will lead to business
cycle comovement. The foreign economy experiences a downturn in both the output gap
and inflation during the period in which the natural rate is diminished. Qualitatively, this is
more intense when natural rate spillovers are relatively large (i.e. when home bias is small).
Quantitatively, the difference seems to be not too large. However, the optimal cooperative
fiscal gap is qualitatively different according to the degree of home bias. When there is a low
degree of home bias and demand spillovers are large, then the optimal fiscal policy response
is to increase government spending along with the more severely impacted home economy.
However, at greater degrees of home bias as the foreign economy is more insulated from
the demand shock a, the optimal policy will be to cut government spending in the foreign
economy in order to concentrate it on the home economy. As the economy experiences
more exchange rate appreciation, they will modify the fiscal policy in a more expansionary
direction. At the Benchmark, the initial response is to cut foreign fiscal spending.
The response of foreign fiscal policy depends on the effectiveness of fiscal policy in overcoming the zero lower bound. We examine the response of the home and foreign economies
to a home demand shock at the Benchmark parameterization and at a unit elasticity of
intertemporal substitution,  = 1. At the unit elasticity of substitution, the global multiplier


is approximately 3 as opposed to 1.5 at the benchmark level. Persistent fiscal policy
will reduce real interest rates which will have a more stimulative effect on demand when
demand is interest elastic. The optimal aggregate fiscal policy will be very large when fiscal
policy is very effective. In this case, the optimal cooperative policy will be for the both the
home and the foreign government to expand spending, though this will still be concentrated
more in the home economy. Panel C shows that optimal discretionary policy of the foreign
economy. When demand is sufficiently intertemporally elastic, both countries will optimally
and persistently expand. At this elasticity, the economy is unstable and the output gap,
inflation, and consumption
23
8
Conclusion
When a large negative demand shock constrains policy interest rates at the zero lower bound,
optimal fiscal policy will be expansionary. When home bias concentrates the demand shock
within a region, optimal policy will also be concentrated in that region. If the aggregate
expansion is sufficiently large and the degree of concentration is sufficiently small, then both
regions should expand fiscal spending. This might typically be characterized by a large
aggregate contraction with relatively small regional differentials. On the other hand, if the
degree of concentration is sufficiently intense as in teh case of sufficient home bias in spending,
fiscal policy amongst regions should diverge. This may be the case even if trade and financial
channels cause both countries to experience deflatonary contractions.
The above analysis assumes that both countries have unconstrained fiscal policy. If a less
exposed region of the currency union was the only region with the fiscal space to implement
fiscal policy, it might need towar direct spending to the aggregate decline at the cost of
exacerbating regional differences. We will pursue this in future work.
9
Appendix
9.1
Proofs
Proof for Proposition 1
Proof.


·
=½
2  + 
1−
2
Ξ
+ (1 − ) +
√
Ξ+
2
[Ξ+{1+}] −4
2
¾·

 + 
q
This is [Ξ + {1 + }] −4 = [Ξ + {1 − }] +4  (Ξ) so Ξ+ [Ξ + {1 + }]2 − 4  2×Ξ.
2
2
In absolute terms, the numerator of
So 0    = [1 −


]
2 +

2
 1.
2
is larger than the denominator, 1 − z =

+
 1
Proof for Proposition 2

Proof. We write 2 · +
= 2 · [1 − z ], so   = 1 − 2 · [1 − z ] = [1 − 2 ] + 2 · z .

q
Define (Ξ) ≡ [Ξ + {1 + }]2 − 4 where  0 (Ξ) = Ξ+{1+}
 0 Also define  = 1 −  +
(Ξ)
2(1 − )  0. We can write

2
· z =

{Ξ+1−+2(1−)+(Ξ)}
24
is a declining function of Ξ. We
can write
[1 −

2Ξ
] = (Ξ) = 1 −
2
{Ξ +  + (Ξ)}
 + (Ξ) − Ξ
=
{Ξ +  + (Ξ)}
So that
( 0 (Ξ) − 1) {Ξ +  + (Ξ)} − (0 (Ξ) + 1) { + (Ξ) − Ξ}
{Ξ +  + (Ξ)}2
( 0 (Ξ)) [{Ξ +  + (Ξ)} − { + (Ξ) − Ξ}] + [{Ξ −  − (Ξ)} − {Ξ +  + (Ξ)}]
=
{Ξ +  + (Ξ)}2
( 0 (Ξ)2Ξ) − 2 { + (Ξ)}
(Ξ) (0 (Ξ) − { + (Ξ)}
=
=
2
(Ξ) {Ξ +  + (Ξ)}2
{Ξ +  + (Ξ)}2
((Ξ) 0 (Ξ)Ξ − (Ξ) { + (Ξ)}
[(Ξ) 0 (Ξ)Ξ − (Ξ)2 ] − (Ξ)
= 2
=
2
(Ξ) {Ξ +  + (Ξ)}2
(Ξ) {Ξ +  + (Ξ)}2
0 (Ξ) =
Note (Ξ) 0 (Ξ)Ξ = Ξ2 + {1 + }Ξ, and (Ξ)2 = [Ξ2 + 2{1 + }Ξ] + (1 − )2 ] which implies
(Ξ) 0 (Ξ)Ξ − (Ξ)2 = −{1 + }Ξ − (1 − )2
0 (Ξ) = 2
So   is declining in Ξ =
−{1 + }Ξ − (1 − )2 − (Ξ)
0
(Ξ) {Ξ +  + (Ξ)}2

2z 
 +
= 2 +
, and thus declining in
2 

  so will be larger when home bias
=
declining in  when 1   ≤ 2, then
. Since  is
is larger.
Proof for Proposition 3
Proof. We can write  () =  ( − 1) −  · 2 ·    ( − 1). For any ,
h
i

1 


 ()   (∞) = 1 − 1− 2 · + . We can write
q
[Ξ + {1 + }] − [Ξ + {1 + }]2 − 4
2
−
2
2
¶
µq
1
2
[Ξ + {1 − }] + 4Ξ − [Ξ + {1 − }]
=
2
1− =
25
and

2
= ½
Ξ
Ξ
2
+
1−
2
+ (1 − ) +
√
[Ξ+{1+}]2 −4
2
¾
2Ξ
¾
= ½
q
2
[Ξ + {1 − }] + 2(1 − ) + [Ξ + {1 − }] + 4Ξ
Define
 = [Ξ + {1 − }]
=
p
 2 + 4Ξ  = 2(1 − )
1
2
2Ξ

=
=
1−
 −
2
 + +
1 
4Ξ
4Ξ
=
=
1− 2
( −  )( +  + )
( −  )( +  ) + ( −  )
4Ξ
4Ξ
4Ξ
=
=
=
2(1−)
2
2
( −  ) + ( −  )
4Ξ + ( −  )
4Ξ +
( −  )

Ξ
4Ξ
=
1
=
4Ξ + 4(1 − )(1 − )
Ξ + (1 − )(1 − )
So
1 
1− 2
[1 − z ]  1
Proof for Proposition 4

Proof. The zero bound multiplier 
exceeds one for  ≤  and is zero for   .

The differentials multiplier  ()  1 for  ≤  and is zero for   . So 
()   ()
for all 
Proof for Proposition 5
Proof. If monetary policy implements zero inflaiton in every period (i.e.  → ∞), the
aggregage multiplier  = z. The long run multiplier is  (∞) = (1 −
£
¤
1 
1 
1 
1 − 1−
·
(1
−
z
)
= 1−
·  + (1 − 1−
). Calculate1 − 2

2
2
2
1 
1− 2
1 
1−
1− 2
and  (∞) =
1 
1− 2
z + (1 −
 (∞) =
1 
1− 2
1 
1− 2
Ξ
Ξ + (1 − )(1 − )
(1 − )(1 − )
=
Ξ + (1 − )(1 − )
=
)
1 
1 
Ξz + (1 − )(1 − )
z + (1 −
)=
1− 2
1− 2
Ξ + (1 − )(1 − )
26
·

)
+
=
Rewrite  = z
 = z =
Calculate
Ξz + z(1 − )(1 − )
Ξ + (1 − )(1 − )
Ξ(z − z ) + (z − 1)(1 − )(1 − )
Ξ + (1 − )(1 − )
q
At  = 2  = 1 and z = z  Since [Ξ + {1 + }]2 − 4  Ξ and (z − 1)  0, if  = 2
 −  (∞) =
then  −  (∞)  0. By continuity, there exists   2 where  −  (∞)  0.
Since (∞)   () for any finite  then  −  ()  0.
Proof for Proposition 6
Proof. The impact effect is a function of Φ

³←
→´
Ξ
(1 + Φ)
=
h
1−
←
→i

2
(1 + Φ)
where  () is defined as in the proof of Proposition 2, 0 ()  0 The impact multiplier is a
function of Φ whose derivative is written as
³ →´
³ →´ ←
³ →´ ←
³←
³←
³←
→ 1−z
→
→´
→´
→´
z  −1
0 ←
0 ←
0 ←
− Ξ Ξ (1+z Φ) −  Ξ
− Ξ Ξ −  Ξ
 Ξ (1+Φ)2 Ξ (1 + Φ) −  Ξ
=

(1 + Φ)2
(1 + Φ)2
(1 + Φ)2
³←
³ →´ ←
→´ ←
→
→ 1−z
0 ←
Ξ
Ξ

−
Ξ Ξ (1+z
where the last inequality follows since −
 0. From the
 Φ)
0
proof of Proposition 2
[1 −
 + (Ξ) − Ξ
(Ξ)2 − Ξ2 +  2 + 2(Ξ)

] = (Ξ) =
=
2
{Ξ +  + (Ξ)}
{Ξ +  + (Ξ)}2
q
where  ≡ 1 −  + 2(1 − ) and (Ξ) ≡ [Ξ + {1 − }]2 + 4Ξ so
(Ξ)2 = [Ξ + {1 − }]2 + 4Ξ = Ξ2 + [2{1 − } + 4] Ξ + {1 − }2
£
¤
(Ξ)2 − Ξ2 +  2 + 2(Ξ) = [2{1 − } + 4] Ξ + {1 − }2 +  2 + 2(Ξ) =
¤
£
[2 + 4] Ξ + {1 − }2 +  2 + 2(Ξ)
From the proof of Proposition2, 0 (Ξ) = 2
−0 (Ξ)Ξ =
[0 (Ξ)Ξ−(Ξ)2 ]−
{Ξ++(Ξ)}2
2Ξ + 2(Ξ)Ξ − 2 0 (Ξ)Ξ2
{Ξ +  + (Ξ)}2
27
So we can write
−0 (Ξ)Ξ − (Ξ) =
2Ξ + 2(Ξ)Ξ − 2 0 (Ξ)Ξ2
−
{Ξ +  + (Ξ)}2
[2 + 4] Ξ + [{1 − }2 +  2 ] + 2(Ξ) 2
{Ξ +  + (Ξ)}2
Calculating
2(Ξ)(Ξ − ) − 20 (Ξ)Ξ2 − 4Ξ − [{1 − }2 +  2 ]
{Ξ +  + (Ξ)}2
2(Ξ)(Ξ − ) − 20 (Ξ)Ξ2 − 4Ξ

{Ξ +  + (Ξ)}2
2(Ξ)2 (Ξ − ) − 2(Ξ)0 (Ξ)Ξ2 − 4Ξ(Ξ)
=
(Ξ) {Ξ +  + (Ξ)}2
=
Note (Ξ)0 (Ξ)Ξ = Ξ2 + {1 + }Ξ, and (Ξ)2 = [Ξ2 + 2{1 + }Ξ] + (1 − )2 ] which implies
(Ξ)2 (Ξ − ) − (Ξ) 0 (Ξ)Ξ2
£
¤
= Ξ3 + 2{1 + }Ξ2 + (1 − )2 Ξ] − Ξ3 − {1 + }Ξ2
£
¤
− Ξ2 + 2{1 + }Ξ + (1 − )2 ]
£
¤
= {1 + }Ξ2 + (1 − )2 Ξ −  Ξ2 + 2{1 + }Ξ + (1 − )2 ]
£
¤
= [{1 + } − ]Ξ2 + (1 − )2 − 2{1 + } Ξ − (1 − )2
We know [{1 + } − ] = [{1 + } − 1 −  − 2(1 − )] = 2 and (1 − )2 − 2{1 + } =
(1 − )[(1 − ) − 2(1 + ) − 2(1 − )] = −(1 − )[(1 + 3 + 2(1 − )]  0, so
2(Ξ)2 (Ξ − ) − 2(Ξ) 0 (Ξ)Ξ2 − 4Ξ(Ξ)
(Ξ) {Ξ +  + (Ξ)}2
4 [Ξ2 − Ξ(Ξ)] − (1 − )[(1 + 3 + 2(1 − )]Ξ − (1 − )2
=
(Ξ) {Ξ +  + (Ξ)}2
 −0 (Ξ)Ξ − (Ξ)
Since(Ξ)  Ξ, then[Ξ2 − Ξ(Ξ)]  0 so all of the elements are less than zero. Thus, the
impact effect of demand shocks is a declining function of Φ whenΦ  0.
28
9.2
Parameter Derivations
Here we define the parameters used in the loss function, which is taken from that used in
Cook and Devereux (2011).
 ≡
 ≡
 ≡
 ≡
 ≡
 ≡
½
¾
(1 − 2 )
(1 +  ) ( − )
( + )
(1 +
) =
+
2
2


 

( − 1)(1 − 2 ) () 1 + 2 ( − 1)
+
(
)  
 

 
( +  )
( + )
=

2

1


1
−

=

≡
=
−
(1 −  ) 2
(1 −  ) 
2

∙
¸
1
( − )
− 2 − 2 2 (1 + ( − 1)( − 1)2 )

 
((1 −  ) +  ) ( − )
+ 2 2 (1 + ( − 1)( − 1)2 )
(1 −  )2
 
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32
(A) Natural Rate
(B) Multiplier
4
-200
% Deviation
Annualized Basis Points
0
-400
-600
-800
-1000
0
2
4
6
8
10
3
2
1
0
12
0
2
0
3
-2
2
1
0
0
2
4
6
8
10
-8
12
-5
%
% Deviation
-1
-2
4
6
12
0
2
4
6
8
10
12
10
12
(F) Consumption
0
2
10
-6
(E) Inflation
0
8
-4
0
-3
6
(D) Output Gap
4
% Deviation
% of GDP
(C) Fiscal Gap
4
8
Low
10
-10
-15
12
Medium
0
2
4
6
8
High Intertemporal Elasticity
Figure 1: Response of Aggregate Economy to Negative Demand Shock
33
(A) Natural Rate
(B) Real Exchange Rate
Annualized Basis Points
0
3
% Deviation
-100
-200
-300
-400
0
5
2
1
0
10
0
(C) Fiscal Gap
% Deviation
% of GDP
1
1
0
0
5
0
-1
-2
-3
10
0
(E) Inflation
5
10
(F) Consumption
0.5
% Deviation
1
%
0
-0.5
-1
10
(D) Output Gap
2
-1
5
0
5
Optimal Policy
0
-1
-2
-3
10
Zero Gap
0
5
10
Counter-cyclical Policy
Figure 2: Response of Regional Differentials to Negative Home Demand Shock Under Alternative Fiscal Policies
34
(B) Real Exchange Rate
4
-200
3
% Deviation
Annualized Basis Points
(A) Natural Rate
0
-400
-600
-800
0
2
4
6
8
10
2
1
0
12
0
2
(C) Fiscal Gap
8
10
12
10
12
10
12
0.5
% Deviation
2
% of GDP
6
(D) Output Gap
3
1
0
0
-0.5
-1
-1
-2
4
0
2
4
6
8
10
-1.5
12
0
2
(E) Inflation
4
6
8
(F) Consumption
0.5
1
% Deviation
0
%
0
-0.5
-1
-2
-3
-1
0
2
4
6
8
Low
10
-4
12
Benchmark
0
2
4
6
8
High Home Bias
Figure 3: Response of Regional Differential Under Different Degrees of Home Bias
35
(A) Natural Rate
(B) Terms of Trade
Annualized Basis Points
0
2.5
2
% Deviation
-100
-200
-300
-400
1.5
1
0.5
0
2
4
6
8
10
0
12
0
2
2
2
1
1
0
-1
0
2
4
6
8
10
10
12
10
12
10
12
-1
0
2
(E) Inflation
4
6
8
(F) Consumption
1
% Deviation
0.2
% Deviation
8
0
-2
12
0.4
0
-0.2
-0.4
-0.6
6
(D) Output Gap
3
% Deviation
% of GDP
(C) Fiscal Gap
4
0
2
4
6
8
Optimal Policy
10
0
-1
-2
-3
12
0
Commitment
2
4
6
8
Counter-cyclical Policy
Figure 4: Response of Regional Differentials to Negative Home Demand Shock Under Optimal Commitment and Discretion
36
(A) Natural Rate
(B) Real Exchange Rate
4
% Deviation
Annualized Basis Points
0
-500
-1000
-1500
0
2
4
6
8
10
3
2
1
0
12
0
2
(C) Fiscal Gap
4
6
8
10
12
10
12
10
12
(D) Output Gap
4
1
% Deviation
% of GDP
0
2
0
-1
-2
-3
-2
0
2
4
6
8
10
-4
12
0
2
(E) Inflation
8
2
0
% Deviation
0
%
6
(F) Consumption
1
-1
-2
-3
4
-2
-4
-6
0
2
4
6
8
Low
10
-8
12
Benchmark
0
2
4
6
8
High Home Bias
Figure 5: Response of the Home Economy to Negative Home Demand Shock
37
(A) Natural Rate
(B) Real Exchange Rate
Annualized Basis Points
0
0
% Deviation
-200
-400
-600
-800
0
2
4
6
8
10
-1
-2
-3
-4
12
0
2
1
1
0
0
-1
-2
0
2
4
6
8
10
10
12
10
12
10
12
-2
0
2
(E) Inflation
4
6
8
(F) Consumption
0
% Deviation
-0.5
%
8
-1
-3
12
0
-1
-1.5
-2
6
(D) Output Gap
2
% Deviation
% of GDP
(C) Fiscal Gap
4
0
2
4
6
8
10
Low
-1
-2
-3
12
0
Benchmark
2
4
6
8
High Home Bias
Figure 6: Response of the Foreign Economy to Negative Demand Shock
38
(A) Natural Rate
(B) Real Exchange Rate
0
-100
% Deviation
Annualized Basis Points
0
-200
-300
-1
-2
-400
-500
0
2
4
6
8
10
-3
12
0
2
2
1.5
0
1
0.5
0
-0.5
0
6
8
10
12
10
12
10
12
(D) Output Gap
2
% Deviation
% of GDP
(C) Fiscal Gap
4
-2
-4
-6
2
4
6
8
10
-8
12
0
2
(E) Inflation
4
6
8
(F) Consumption
0
2
% Deviation
0
%
-1
-2
-2
-4
-6
-3
0
2
4
6
8
10
Unit Elasticity
-8
12
0
Benchmark
2
4
6
8
Low Elasticity
Figure 7: Foreign Response to Negative Demand Shock Under Intertemporal Elasticity
39