xxx

Real Rigidity, Nominal Rigidity,

and the Social Value of Information

by George-Marios Angeletos, Luigi Iovino and Jennifer La’O1

Online Appendices

August 19, 2015

This document contains the two online appendices for our article.

Appendix A: Proofs for the Baseline Model

This appendix contains the proofs of all the formal results that appear in Sections II-III of our paper.

(The numbering of equations, lemmas, and propositions throughout this document is consistent

with the one used in the paper.)

Derivation of equation (1). Let pit

be the price index for the consumption basket of the

goods produced in island i. The optimal consumption decision satisfies

cit

=

✓

pit

Pt

◆�⇢

Ct

and cijt

=

✓

pijt

pit

◆�⌘

it

cit

,

for, respectively, the aforementioned basket and the particular good produced by firm j in island i.

In equilibrium, consumption coincides with production. It follows that the inverse demand function

faced by firm j in island i is given by

pijt

= Dit

y� 1

⌘

it

ijt

, (14)

where Dit

⌘ pit

y1

⌘

it

it

= Pt

Y1

⇢

t

y1

⌘

it

�1

⇢

it

is taken as given by the individual firm but is determined

endogenously within the island.

1Angeletos: Department of Economics, MIT, and NBER, angelet@mit.edu; Iovino: Department of Economics,

Bocconi University, and IGIER, luigi.iovino@unibocconi.it; La’O: Department of Economics, Columbia University,

and NBER, jenlao@columbia.edu.

1

Consider now the optimal behavior of the individual firm. Given that the marginal value of

(nominal) income for the representative household is U 0(Yt

)/Pt

, the firm’s objective is simply the

local expectation of its profit times U 0(Yt

)/Pt

. Using (14), this can be expressed as follows:

Eit

U 0 (Yt

)

Pt

✓

Dit

y1� 1

⌘

it

ijt

� wit

nijt

◆�

Using yijt

= Ai

nijt

and taking the FOC with respect to nijt

gives

Eit

2

6

4

✓

1� 1

⌘it

◆

Ait

U 0 (Yt

)D

it

y� 1

⌘

it

ijt

Pt

� U 0 (Yt

)wit

Pt

3

7

5

= 0.

By the fact that all firms within a given island are symmetric, we have that, in equilibrium,

nijt

= nit

, yijt

= yit

, and pijt

= pit

. It follows that Dit

y� 1

⌘

it

ijt

= Pt

Y1

⇢

t

y�1

⇢

it

and the above condition

reduces to

Eit

U 0 (Yt

)wit

Pt

�

= Eit

"

✓

1� 1

⌘it

◆

U 0 (Yt

)Y1

⇢

t

y�1

⇢

it

Ait

#

Finally, consider the optimal labor supply in island i. The relevant FOC for the household is

�it

V 0 (nit

) = (1� ⌧it

)Eit

U 0 (Yt

)wit

Pt

�

Combining the above two conditions and letting Mit

⌘ 1

1�⌧

it

⌘

it

⌘

it

�1

gives condition (1). ⌅

Proof of Lemma 1. Taking logs of both sides of (1) and rearranging gives us

⇣

1

⇢

+ ✏⌘

log yit

= �µit

+ logEit

"

Y1

⇢

��

t

#

+ (1 + ✏) ait

.

Assuming Yt

is log-normal (which we verify below), the latter can be rewritten as

log yit

= �0

+ �a

ait

+ �µ

µit

+ ↵Eit

[log Yt

] ,

where

�0

⌘ 1

2⇢

1+⇢✏

⇣

1

⇢

� �⌘

2

V ar (log Yt

) , �a

⌘ ⇢(1+✏)

1+⇢✏

, �µ

⌘ � ⇢

1+⇢✏

, ↵ ⌘ 1�⇢�

1+⇢✏

.

Note that �a

> 0 and �µ

< 0, reflecting the fact that local output increases with local productivity

and decreases with the local level of monopoly power. Finally, ↵ could be either positive or negative,

but it is necessarily less than 1. ⌅

2

Proof of Lemma 2. Welfare is given by

W =X

�tWt

where

Wt

⌘ E"

Y 1��

t

1� �� 1

1+✏

Z

�it

✓

yit

Ait

◆

1+✏

di

#

measures the unconditional expectation of the welfare flow in period t. Because the aggregate shocks

are i.i.d. across time and all second moments are time-invariant,2 the unconditional expectations of

all the objects that enter into Wt

are time-invariant, and hence Wt

is itself a time-invariant function

of the underlying preference, technology, and information parameters. To simplify the notation, we

thus drop the time index t in the rest of this proof and proceed to develop a certain decomposition

of the welfare flow W for an arbitrary period.

Before doing this, we highlight a property of log-normal distributions that is utilized repeatedly

in this appendix. When a variable X is log-normal with lnX ⇠ N�

x,�2�

, then, for any � 2 R, wehave that

E[X�] = exp�

�x+ 1

2

�2�2�

=�

exp�

x+ 1

2

�2��

�

exp�

1

2

(� � 1)��2�

and therefore

E[X�] = (E[X])� exp�

1

2

(� � 1)��2�

. (15)

We use this property again and again in the derivations that follow, for various X and �.

Consider the first component ofW , which corresponds to the utility of consumption and which is

given by 1

1��

E�

Y 1��

�

. Noting that equilibrium Y is log-normal and using the log-normal property,

we have that

E�

Y 1��

�

= [E (Y )]1�� exp�

�1

2

� (1� �)V ar (log Y )

(16)

Consider now the second component of W , which corresponds to the disutility of labor. Defining

bi

⌘ A1+✏

i

/�i

, letting B denote the cross-sectional mean of bi

, noting that Y =

✓

R

y⇢�1⇢

i

di

◆

⇢

⇢�1

,

and using once again the log-normal property, we can express the realized disutility of labor, in any

given state, as follows:

Z

�i

✓

yi

Ai

◆

1+✏

di = E

Z

y1+✏

i

bi

�

=Y 1+✏

Bexp(H)

where

H ⌘ 1

2

⇣

✏+ 1

⇢

⌘

(1 + ✏)V ar (log yi

|⇥) + 1

2

V ar (log bi

|⇥)� (1 + ✏)Cov (log yi

, log bi

|⇥)

2These assumptions are for expositional simplicity; otherwise, the welfare results we document would have to be

restated simply by distinguishing the information structure period by period.

3

and where ⇥ ⌘ (Y,B) encapsulates the aggregate state of the economy. It follows that the expected

disutility of labor is given by

E"

Z

�i

✓

yi

Ai

◆

1+✏

di

#

= E

Y 1+✏

B

�

exp(H) =E[Y ]1+✏

E[B]exp(G) (17)

where we have used once again the property from (15) to obtain

G ⌘ H + 1

2

✏(1 + ✏)V ar (log Y ) + V ar (logB)� (1 + ✏)Cov (log Y, logB) .

Because of our Gaussian specification, the variance and covariance terms that enter H and G above

are constants (non-random and time-invariant), and hence H and G are themselves constants.

Combining (16) and (17), we infer that the per-period welfare is given by

W = 1

1��

[E (Y )]1�� exp�

�1

2

� (1� �)V ar (log Y )

� 1

1+✏

[E(Y )]

1+✏

E(B)

exp(G). (18)

Next, let us define Y as the value of E (Y ) that maximizes expression (18) for W , taking as given

B,G, and V ar (log Y ). Clearly, this is given by taking the FOC of (18) with respect to E(Y ) and

equating this with 0, or equivalently by the solution to the following condition:

Y 1�� exp�

�1

2

� (1� �)V ar (log Y )

=ˆ

Y

1+✏

E(B)

exp(G) (19)

We can then restate W as follows:

W =

(

1

1��

E (Y )

Y

�

1��

� 1

1+✏

E (Y )

Y

�

1+✏

)

ˆ

Y

1+✏

E(B)

exp(G)

If E(Y ) happens to equal Y , then W = W , where

W ⌘ ✏+�

(1��)(1+✏)

ˆ

Y

1+✏

E(B)

exp(G). (20)

Letting

� ⌘ E(Y )

Yand v(�) ⌘ U(�)� V (�)

U(1)� V (1)=

1

1��

�1�� � 1

1+✏

�1+✏

✏+�

(1��)(1+✏)

,

we conclude that

W = v(�)W . (21)

The term v(�) therefore identifies the wedge between actual welfare, W , and the reference level

W that a planner could have a↵orded if he had a non-contingent subsidy that permitted him to

scale up and down the mean level of output and could use it to maximize welfare. To see this more

clearly, note that v(�) is strictly concave in � and reaches its maximum at � = 1 when � < 1,

whereas it is strictly convex and reaches its minimum at � = 1 when � > 1. Along with the fact

that W > 0 when � < 1 but W < 0 when � > 1 (this fact will be come clear momentarily), this

means that Wv(�) is always strictly concave in �, with the maximum attained at � = 1.

4

So far, we have decomposed the per-period welfare flow as W = Wv(�). In what follows, we

proceed to decompose the reference level W itself into the product of two terms: the first-best level

W ⇤; and a function of ⇤, which encapsulates the welfare losses of volatility and dispersion.

From (19), we have that

Y = [E (B)]1

✏+� expn

� 1

✏+�

⇥

G+ 1

2

� (1� �)V ar (log Y )⇤

o

,

which together with (20) gives

W = ✏+�

(1��)(1+✏)

[E (B)]1��

✏+� expn

G� 1+✏

✏+�

⇥

G+ 1

2

� (1� �)V ar (log Y )⇤

o

Equivalently,

W = ✏+�

(1��)(1+✏)

[E (B)]1��

✏+� expn

�1

2

(1��)(1+✏)

✏+�

⌦)o

(22)

where

⌦ ⌘ 2

1+✏

G+ �V ar (log Y )

= (✏+ �)V ar (log Y ) + 2

1+✏

V ar (logB)� 2Cov (log Y, logB)

+⇣

✏+ 1

⇢

⌘

V ar (log yi

|⇥) + 1

1+✏

V ar (log bi

|⇥)� 2Cov (log yi

, log bi

|⇥)

Now, note that the first-best levels of output are given by the fixed point to the following equation:

log y⇤i

= (1� ↵) 1

✏+�

log bi

+ ↵ log Y ⇤.

It follows that, up to some constants that we omit for notational simplicity,

log Y ⇤ = 1

✏+�

logB and log y⇤i

� log Y ⇤ = (1� ↵) 1

✏+�

(log bi

� logB)

Using this result towards replacing the terms in ⌦ that involve bi

and B, we get

⌦ = (✏+ �)V ar (log Y ) + 2 (✏+�)

2

(1+✏)

V ar (log Y ⇤)� 2(✏+ �)Cov (log Y, log Y ⇤)

+⇣

✏+ 1

⇢

⌘

V ar (log yi

|⇥) + (✏+�)

2

(1+✏)(1�↵)

2V ar (log y⇤i

|⇥)� 2 ✏+�

1�↵

Cov (log yi

, log y⇤i

|⇥)

Furthermore, the first-best level of welfare is given by

W ⇤ = ✏+�

(1��)(1+✏)

[E (B)]1��

✏+� expn

�1

2

(1��)(1+✏)

✏+�

⌦⇤)o

(23)

where ⌦⇤ obtains from ⌦ once we replace yi

and Y with, respectively, y⇤i

and Y ⇤ (which have

themselves been obtained above as functions of the exogenous objects bi

and B). We conclude that

W = W ⇤ expn

�1

2

(1��)(1+✏)

✏+�

⇣

⌦� ⌦⇤⌘o

(24)

5

Finally, using the definitions of ⌦ and ⌦⇤ together with 1� ↵ = ✏+�

✏+1/⇢

, we have

⌦� ⌦⇤

✏+ �= {V ar (log Y ) + V ar (log Y ⇤)� 2Cov (log Y, log Y ⇤)}

+1

1� ↵{V ar (log y

i

|⇥) + V ar (log y⇤i

|⇥)� 2Cov (log yi

, log y⇤i

|⇥)}

= V ar (log Y � log Y ⇤) +1

1� ↵V ar (log y

i

� log y⇤i

|⇥)

Note that conditioning on ⇥ ⌘ (log Y, logB) is equivalent to conditioning on (log Y, log Y ⇤). Fur-

thermore, because log Y and log Y ⇤ are the cross-sectional means (expectations) of, respectively,

log yi

and log y⇤i

, we have that

V ar (log yi

� log y⇤i

— log Y, log Y ⇤) =

= V ar ((log yi

� log Y )� (log y⇤i

� log Y ⇤)— log Y, log Y ⇤)

= V ar⇣

log eyi

� log eY⌘

Combining the above results with the definitions of ⌃, � and ⇤, yields

⌦� ⌦⇤

✏+ �= ⌃+

1

1� ↵� = ⇤,

and therefore (24) can be restated as

W = W ⇤ exp�

�1

2

(1 + ✏)(1� �)⇤

, (25)

which gives the sought-after decomposition of W .

Note from (23) that the sign of W ⇤ is the same as the sign of (1� �). It follows that the sign

of W is also the same as that of (1� �), which in turn verifies the claim made earlier on that the

product Wv(�) is strictly convex in � with a maximum value of 1 attained at � = 1.

Finally, combining (25) with (21), we conclude that

W = v(�)w(⇤)

where w(x) ⌘ W⇤ exp�

�1

2

(1 + ✏)(1� �)x

for every x and where W⇤ ⌘ 1

1��

W ⇤ is the first-best

level of (life-time) welfare. The proof is then completed by noting once again that W ⇤ has the same

sign as 1� � and therefore that w is a strictly decreasing function of ⇤, regardless of whether � is

greater or smaller than 1. The fact that W is strictly concave in �, with a maximum attained at

� = 1, follows directly from our earlier observation that W = v(�)W has these exact properties.

⌅

6

Equilibrium with productivity shocks. Suppose the equilibrium production strategy takes a

log-linear form:

log yit

= '0

+ 'a

ait

+ 'x

xit

+ 'z

zt

, (26)

for some coe�cients ('a

,'x

,'z

). Aggregate output is then given by

log Yt

= '0

+X + ('a

+ 'x

) at

+ 'z

zt

where

X ⌘ 1

2

✓

⇢� 1

⇢

◆

V ar(log yit

|⇥) = 1

2

✓

⇢� 1

⇢

◆

'2

a

⇠

+'2

x

x

+ 2'a

'x

x

�

adjusts for the curvature in the CES aggregator. It follows that Yt

is log-normal, with

Eit

[log Yt

] = '0

+X + ('a

+ 'x

)Eit

[at

] + 'z

zt

(27)

V arit

[log Yt

] = ('a

+ 'x

)2 V arit

[at

] (28)

where, by standard Gaussian updating,

Eit

[at

] =

x

a

+

x

+

z

xit

+

z

a

+

x

+

z

zt

(29)

V arit

[at

] = 1

a

+

x

+

z

(30)

Because of the log-normality of Yt

, the fixed-point condition (1) reduces to following:

log yit

= (1� ↵)( ait

� 0 log M) + ↵Eit

[log Yt

] + � (31)

where ⌘ 1+✏

✏+�

> 0, 0 ⌘ 1

✏+�

> 0, log M ⌘ � logh⇣

⌘�1

⌘

⌘

(1� ⌧)i

⇡ µ + ⌧ > 0 is the overall

distortion caused by the monopoly markup and the labor wedge (which are both constant because

we are herein focusing on the case with only productivity shocks), and

� = 1

2

↵⇣

1

⇢

� �⌘

V arit

[log Yt

] = 1

2

↵2

⇣

1

⇢

+ ✏⌘

V arit

[log Yt

] > 0

Next, combining (31) with (27) and (29), we obtain

log yit

= �� (1� ↵) 0�s

+ (1� ↵) ait

+ ↵ ('0

+X + 'z

zt

)

+↵ ('a

+ 'x

)⇣

x

a

+

x

+

z

xit

+

z

a

+

x

+

z

zt

⌘

For this to coincide with our initial guess in (26) for every realization of shocks and signals, it is

necessary and su�cient that the coe�cients ('0

,'a

,'x

,'z

) solve the following system:

'0

= �� (1� ↵) 0 log M+ ↵('0

+X)

'a

= (1� ↵)

'x

= ↵ ('a

+ 'x

)

x

a

+

x

+

z

'z

= ↵'z

+ ↵ ('a

+ 'x

)

z

a

+

x

+

z

7

The unique solution to this system is given by the following:

'a

= (1� ↵) > 0, 'x

=(1� ↵)

x

a

+ (1� ↵)x

+ z

↵ ,

'z

=

z

a

+(1�↵)

x

+

z

↵ , and '0

= � 0�s

+ 1

1�↵

(↵X + �)

Note then that the coe�cients 'x

and 'z

, which capture the individual response to expectations

of the aggregate state, are positive if and only if ↵ > 0. ⌅

Proof of Proposition 1. Using the characterization of the equilibrium allocation in the preceding

proof along with that of the first best in the proof of Lemma 2, we can calculate the equilibrium

value of the aggregate and local output gaps as follows:

log Yt

� log Y ⇤t

= ('a

+ 'x

+ 'z

) at

+ 'z

"t

� at

log yit

� log y⇤it

= 'x

uit

It follows that the volatility of the aggregate output gap is

⌃ ='2

z

z

+('

a

+ 'x

+ 'z

� )2

a

=↵2 (

a

+ z

)

((1� ↵)x

+ z

+ a

)2 2

and the cross-sectional dispersion of the local output gaps is

� ='2

x

x

=↵2 (1� ↵)2

x

((1� ↵)x

+ z

+ a

)2 2.

Taking the derivative of ⌃ with respect to the precision of public information gives

@⌃

@z

=(1� ↵)

x

� (a

+ z

)

((1� ↵)x

+ z

+ a

)3↵2 2

which is negative if and only if z

> (1� ↵)x

� a

, while taking the derivative of � gives

@�

@z

= �2↵2 (1� ↵)2

x

((1� ↵)x

+ z

+ a

)3 2

which is necessarily negative.

Similarly, taking the derivatives of ⌃ and � with respect to the precision of private information,

we obtain@⌃

@x

= � 2 (1� ↵) (a

+ z

)

((1� ↵)x

+ z

+ a

)3↵2

�

0�2

which is necessarily negative and

@�

@x

=

z

+

a

�(1�↵)

x

((1�↵)

x

+

z

+

a

)

3 (1� ↵)2 ↵2

�

0�2

which is negative if and only if (1� ↵)x

> z

+ a

. ⌅

8

Proof of Theorem 1. From the proof of Proposition 1, we can rewrite ⇤ as

⇤ = ⌃+1

1� ↵� =

↵2

((1� ↵)x

+ z

+ a

) 2

from which it is immediate that ⇤ is decreasing in the precision of either public or private infor-

mation, regardless of the sign of ↵. Furthermore,

@2⇤

@z

@↵= � 2↵ (

x

+ z

+ a

)

((1� ↵)x

+ z

+ a

)3 2

which is itself negative if and only if ↵ > 0. Finally, note that the distortion in the mean level of

output is given by

� = M1

✏+� ⌘h⇣

⌘�1

⌘

⌘

(1� ⌧)i

1

✏+�

< 1

where ⌘�1

⌘

is the monopoly wedge (the reciprocal of the markup) and 1 � ⌧ is the labor wedge.

Since � is invariant to the information structure, the welfare e↵ects of either type of information

are captured by the comparative statics of ⇤ alone, which have been established above. ⌅

Equilibrium with markup shocks. This follows very similar steps as the characterization of

equilibrium in the case with productivity shocks. Suppose equilibrium output takes a log-linear

form:

log yit

= '0

+ 'µ

µit

+ 'x

xit

+ 'z

zt

,

for some coe�cients ('µ

,'x

,'z

). This guarantees that aggregate output is log-normal, which in

turn implies that the fixed-point condition (1) now reduces to

log yit

= (1� ↵)( a� 0µit

) + ↵Eit

[log Yt

] + �

where , 0, and � are defined as in the case with productivity shocks. Following similar steps as

in that case, we can then show that the unique equilibrium coe�cients are given by the following:

'µ

= � (1� ↵) 0 < 0, 'x

= � (1�↵)

x

µ

+(1�↵)

x

+

z

↵ 0,

'z

= �

z

µ

+(1�↵)

x

+

z

↵ 0, and '0

= a+ 1

1�↵

(↵X + �)

Note that the sign of the coe�cients 'x

and 'z

is once again pinned down by the sign of ↵. ⌅

Proof of Proposition 2. With only markup shocks, the first-best levels of output are constant.

The volatility of aggregate output gaps and the dispersion of local output gaps are thus given by

the following:

⌃ ='2

z

z

+('

µ

+ 'x

+ 'z

)2

µ

=↵2

µ

z

+ ((1� ↵)µ

+ (1� ↵)x

+ z

)2

µ

(µ

+ (1� ↵)x

+ z

)2�

0�2 (32)

9

� ='2

µ

⇠

+'2

x

x

+ 2'µ

'x

x

=(1� ↵)2

⇠

�

0�2 +↵ (1� ↵)2 (2

µ

+ (2� ↵)x

+ 2z

)

((1� ↵)x

+ z

+ µ

)2�

0�2 (33)

Next, taking the derivatives with respect to the precision of public information, we obtain

@⌃

@z

=(2 + ↵) (1� ↵)

x

+ (z

+ µ

) (2� ↵)

((1� ↵)x

+ z

+ µ

)3↵�

0�2

which is positive if ↵ > 0 and

@�

@z

= �2 (1� ↵)2 (µ

+ x

+ z

)

((1� ↵)x

+ z

+ µ

)3↵�

0�2

which is negative if (and only if) ↵ > 0.

Similarly, taking the derivatives of ⌃ and � with respect to the precision of private information,

we obtain@⌃

@x

=2 (1� ↵)2 (

x

+ z

+ µ

)

((1� ↵)x

+ z

+ µ

)3↵�

0�2

which is positive if (and only if) ↵ > 0 and

@�

@x

= �(1� ↵)2 [(2� ↵) (1� ↵)x

+ (2� 3↵) (µ

+ z

)]

((1� ↵)x

+ z

+ µ

)3↵�

0�2 ,

which is in general ambiguous. ⌅

Proof of Proposition 3. Using (32) and (33), we can obtain the equilibrium value of ⇤ as

⇤ =1� ↵

⇠

�

0�2 +(1� ↵)

x

+ z

+ (1� ↵2)µ

µ

((1� ↵)x

+ z

+ µ

)

�

0�2

(Note that the first term captures the distortion caused by cross-sectional dispersion in actual

markups, whereas the second terms captures the distortion caused by the firms’ response to their

information about the aggregate markup.) It follows that, regardless of the sign of ↵,

@⇤

@z

=↵2 ( 0)2

((1� ↵)x

+ z

+ µ

)2> 0 (34)

@⇤

@x

=(1� ↵)↵2 ( 0)2

((1� ↵)x

+ z

+ µ

)2> 0, (35)

which proves that ⇤ increases with the precision of either public or private information, regardless

of the sign of ↵. ⌅

10

Proof of Proposition 4. Recall that � is given by the ratio of the equilibrium value of expected

output, E[Y ], to the corresponding optimal value, Y . The former can be computed from the

preceding equilibrium characterization and the latter from condition (19). After some tedious

algebra (which is available upon request), we can thus show that

� ⌘ E[Y ]

Y= exp

⇥

� 0 �µ+ 1

2

D�⇤

,

where

D ⌘ V ar (µi

) + 2 (1 + ✏)Cov (yi

, µi

)

= 1

⇠

+ 1

µ

+ 2 (1 + ✏)

✓

'µ

⇠

+'x

x

+'µ

+ 'x

+ 'z

µ

◆

= 1

⇠

+ 1

µ

� 2 (1 + ✏)

✓

1� ↵

⇠

+ 1

µ

� ↵2

µ

+ (1� ↵)x

+ z

◆

0

It follows that@�

@z

= �1

2

� 0 @D

@z

=↵2 ( 0)2

((1� ↵)x

+ z

+ µ

)2(1 + ✏)� > 0 (36)

@�

@x

= �1

2

� 0 @D

@x

=(1� ↵)↵2 ( 0)2

((1� ↵)x

+ z

+ µ

)2(1 + ✏)� > 0. (37)

That is, � is increasing in the precision of either public or private information, irrespective of

whether ↵ is positive or negative. ⌅

Proof of Theorem 2. To obtain the overall welfare e↵ect, recall that welfare is given by

W = v(�)w(⇤) = W⇤v(�) exp�

�1

2

(1 + ✏)(1� �)⇤

Consider first the case of public information. From the above, we have that

@W@

z

= W⇤ exp�

�1

2

(1 + ✏)(1� �)⇤

✓

v0(�)@�

@z

� 1

2

v(�)(1 + ✏)(1� �)@⇤

@z

◆

From (34) and (36), we have that@�

@z

=@⇤

@z

(1 + ✏)�

It follows that@W@

z

= W⇤ exp�

�1

2

(1 + ✏)(1� �)⇤ @⇤

@z

H

where

H ⌘ v0(�)(1 + ✏)�� 1

2

(1� �)(1 + ✏)v(�) = (1��)(1+✏)

2(✏+�)

⇥

(1 + ✏)�1�� � (1 + 2✏+ �)�1+✏

⇤

,

and, therefore,

@W@

z

= (1��)(1+✏)

2(✏+�)

W⇤ exp�

�1

2

(1 + ✏)(1� �)⇤ @⇤

@z

�1��

⇥

(1 + ✏)� (1 + 2✏+ �)�✏+�

⇤

11

Note then that the sign of W⇤ is the same as that of (1 � �) which, together with the facts that@⇤

@

z

> 0 and � > 0, implies that the sign of @W@

z

is the same as the sign of (1+ ✏)� (1+2✏+�)�✏+� .

We conclude that@W@

z

< 0 i↵ � > �,

where

� ⌘✓

1 + ✏

1 + 2✏+ �

◆

1

✏+�

2 (0, 1).

Consider next the case of private information. From (35) and (37), we have that

@�

@x

=@⇤

@x

(1 + ✏)�,

as in the case of public information. It follows that

@W@

x

= W⇤ exp�

�1

2

(1 + ✏)(1� �)⇤

✓

v0(�)@�

@x

� 1

2

v(�)(1 + ✏)(1� �)@⇤

@x

◆

= W⇤ exp�

�1

2

(1 + ✏)(1� �)⇤ @⇤

@x

H

where H is defined as before. By direct implication,

@W@

x

< 0 i↵ � > �,

where � is the same threshold as the one in the case of public information.

Clearly, when a non-contingent subsidy is set optimally, E[Y ] = Y or � = 1 > �. ⌅

Appendix B: Auxiliary Results and Proofs for Extended Model

This appendix contains the proofs for Section 4, along with a number of auxiliary results. In Section

B.1, we provide a characterization of the set of implementable allocations, that is, the set of all

allocations that can be part of an equilibrium for some monetary policy; we also show that this set

remains the same whether monetary policy responds to the state within the same period or with

a lag. In Section B.2, we develop a preliminary welfare decomposition, which forms the basis of

the particular decompositions that appear in the main text. In Section B.4, we use a numerical

example to illustrate an argument made in the main text. In Section B.4, we collect the proofs for

all results that appear either in the main text or in Sections B.1 and B.2 of this appendix.

12

B.1 Equilibrium and Implementability

The equilibrium is defined in a similar manner as in the baseline model, modulo the fact that prices

are now set on the basis of incomplete information. Consider the FOCs of firm i, who chooses nit

and pit

so as to maximize the expected valuation of its profit. Combining these conditions with the

household’s FOC for labor supply and for the demand of the di↵erent commodities, we obtain the

following conditions:

0 = n1+✏

it

� Eit

2

4Mit

U 0 (Yt

)

✓

yit

Yt

◆�1

⇢

✓cit

3

5 (38)

0 = Eit

2

4

0

@l1+✏

it

�Mit

U 0 (Yt

)

✓

yit

Yt

◆�1

⇢

⌘yit

1

A

3

5 (39)

where

Yt

=

Z

(qit

l⌘it

)⇢�1⇢ di

�

⇢

⇢�1

; (40)

These conditions are the analogue of condition (1) from the baseline model and identify two of

the four key implementability conditions of the general model. The third condition follows form

the household’s optimal demand for the di↵erent commodities and ties relative prices to relative

quantities:

pit

Pt

=

✓

qit

l⌘it

Yt

◆�1

⇢

; (41)

The last condition follows from the Euler condition of the household and ties the nominal interest

rate to output growth and inflation:

log(1 +Rt

) = const+ � (Et

[log Yt+1

]� log Yt

]) + (Et

[logPt+1

]� logPt

) (42)

To recap, a combination of quantities and prices are part of an equilibrium if and only if (i) the

quantities and prices satisfy conditions (38) through (41) and (ii) monetary policy satisfies (42).

We now proceed to restate these conditions in a manner that facilitates our subsequent analysis.

For expositional purposes, this is done in three steps. First, in Lemma 6, we restrict attention to

the subset of equilibria in which the interest rate is measurable in the current fundamental and

the current public signal. Next, in Lemma 7, we show that exactly the same real outcomes as

those in Lemma 6 obtain if we instead consider the subset of equilibria in which the interest rate

is measurable in the value of the shock at some past period (i.e., if policy reacts with the lag). It

is then immediate that the set of implementable allocations remain the same if we also consider

the more general case in which the interest rate is arbitrary function of the entire history of the

shock. Clearly, the same applies for the public signal. We thus conclude the characterization in

Lemma 8 by considering the residual case in which the interest rate depends also on a shock that

13

is orthogonal to the entire history of the fundamental and the public signal. Throughout, we let

sit

and st

denote the idiosyncratic and aggregate shocks to fundamentals (technology or markups).

Lemma 6 Suppose that the nominal interest rate satisfies

log (1 +Rt

) = ⇢s

st

+ ⇢z

zt

,

for some coe�cients rs

and rz

, and consider the following pair of strategies:3

log qit

= '0

+ 's

sit

+ 'x

xit

+ 'z

zt

and log lit

= l0

+ ls

st

+ ls

sit

+ lx

xit

+ lz

zt

. (43)

(i) When prices are set on the basis of incomplete information, a pair of strategies as in (43)

can be implemented as part of an equilibrium if and only if the following conditions are satisfied:4

's

= �s

(44)

'x

= �x

+ �0x

ls

(45)

'z

= �z

+ �0z

ls

(46)

ls

= 1

✓

('s

� 1s=a

) (47)

lx

= 1

✓

'x

�

x

s

+

x

+

z

ls

(48)

lz

= 1

✓

'z

�

z

s

+

x

+

z

ls

, (49)

where �s

is given in (52), 1s=a

is an indicator that takes the value 1 in the case of technology shocks

(s = a) and 0 in the case of markup shocks (s = µ), �x

, �0x

, �z

, �0z

are scalars given in the proof,

and ls

is an arbitrary coe�cient.

(ii) When instead prices are flexible (i.e., free to adjust to the true sate), there exists a unique

pair of strategies as in (43) that can obtain in equilibrium, and this pair is pinned down by the

combination of conditions (44)-(49) along with the following condition:

ls

= l⇤s

⌘(1� ⇢�)

�

1 + ⌘

✓

�

⇣

�s

+ �x

⌘

� ⌘ (1� ⇢�)1s=a

⇢ (1 + ✏� ⌘) + ⌘ � (1� ⇢�)⇣

�

1 + ⌘

✓

�

�0x

+ ⌘

s

+

z

s

+

x

+

z

⌘ . (50)

This lemma identifies the precise way in which monetary policy can control real allocations.

When prices are set on the basis of incomplete information, by appropriately designing the response

of the interest rate to the realized shock, the policy maker can choose at will the coe�cient ls

,

that is, the response of the second-stage labor input (the margin of adjustment in quantities) to

the realized technology or markup shock. Conditional on choosing this coe�cient, however, the

monetary authority has no further control over the real allocation. In this sense, the coe�cient

3By “strategies” we refer to functions that map the information set of a firm to its employment and production.4There is also a pair of restrictions on '0 and l0, which we omit because these are of no interest: '0 and l0 are

irrelevant for the stochastic properties of the equilibrium.

14

coe�cient ls

is the only “free variable” at the disposal of the policy maker. Finally, when prices

are flexible (free to adjust to the realized shock), this variable ceases to be free, and the policy

maker has, of course, no control over real allocations (although he can still control the nominal

price level).

We now proceed to show that the set of implementable allocations remains the same whether

monetary policy responds to the realized state within the same period or with an arbitrary lag.

Lemma 7 Suppose that the nominal interest rate satisfies

log(1 +Rt

) = ⇢�k

st�k

+ ⇢z

zt

(51)

for some k � 1 and some scalars ⇢�k

and ⇢z

. Parts (i) and (ii) of Lemma 6 continue to hold. That

is, the set of implementable allocations remains the same.

By a similar argument as the one found in the proof of this lemma, the set of implementable

allocations remains the same if we consider the more general class of policies in which the interest

rate is an arbitrary function of the entire history of the fundamental and the public signal. We

thus conclude this section by extending Lemma 6 to the only case that has not been allowed so

far, namely allowing for the interest rate to contain a pure monetary shock, by which we mean a

shock orthogonal to both the fundamental and the public signal (and the histories thereof). This

makes no essential di↵erence to the logic underlying the implementability constraints we derived

in Lemma 6. It only introduces a mechanical response of output to the monetary shock.5

Lemma 8 Suppose that the nominal interest rate satisfies

log (1 +Rt

) = ⇢s

st

+ ⇢z

zt

+ rt

,

where rt

is a Normally distributed random variable that is orthogonal to both st

and zt

and that is

unpredictable by the firms. Then, the second-period labor choice satisfies:

log lit

= l0

+ ls

st

+ ls

sit

+ lx

xit

+ lz

zt

� 1

⌘�rt

.

However, the strategy for qit

remains the same and the implementability conditions (44)-(49) are

also not a↵ected.

B.2 Welfare

In this subsection, we obtain a preliminary welfare decomposition, which extends Lemma 2 from

the baseline model to the more general model under consideration.

5This response would itself be more complicated if firms had information about the monetary shock at the moment

they make their pricing and production decisions, a possibility which we only briefly discuss in the end of Section 5.

15

To this goal, we first introduce certain notation:

✏ ⌘ 1+✏

✓

� 1, � ⌘ 1� (1��)(1+✏)

1+✏�⌘(1��)

, ⇢ ⌘ ⇢(1+✏�⌘)+⌘

1+✏+⌘(1�⇢)

= 1

1�⇢⌫+⌫

,

↵ ⌘ 1�⇢�

1+⇢✏

, � ⌘ 1+✏

1+✏�⌘+�⌘

> 0, ⌫ ⌘ 1+✏

⇢(1+✏�⌘)+⌘

, (52)

�a

⌘ ⇢(1+✏)

1+⇢✏

, �µ

⌘ �⇢+(⇢�1)

⌘

1+✏

1+⇢✏

, ↵ ⌘ (1�⇢�)⌘

⇢(1+✏�⌘)+⌘

.

As in the main text, we also let qit

⌘ Ait

n✓

it

denote the component of output that is fixed on the

basis of the firm’s incomplete information of the state of the economy, and define the corresponding

aggregate as

Qt

⌘

Z

I

(qit

)⇢�1

⇢ di

�

⇢

⇢�1

.

Next, we denote with log yit

and log Yt

the socially optimal levels of, respectively, local and aggregate

output, conditional on an arbitrary allocation of the q’s; and with log q⇤it

and logQ⇤t

the first-best

levels of, respectively, log qit

and logQt

. Finally, we let ⌃Q

and �q

denote, respectively, the volatility

of logQt

� logQ⇤t

and the cross-sectional dispersion of log qit

� log q⇤it

, and similarly we let ⌃Y

and �y

denote, respectively, the volatility of log Yt

� log Yt

and the cross-sectional dispersion of

log yit

� log yit

.

The first of the following two lemmas characterizes the aforementioned reference points, the

first best and the allocation that is optimal conditional on q’s. The second lemma then develops

the desired welfare decomposition in terms of gaps relative to these reference points.

Lemma 9 For any given distribution of q in the cross-section, the optimal output levels solve the

following fixed-point relation:

log yit

= ⇢⌫ log qit

+ ↵ log Yt

. (53)

The first-best allocation satisfies the following fixed-point relation:

log q⇤it

= �a

ait

+ ↵ logQ⇤t

. (54)

Lemma 10 There exists a decreasing function w, which is invariant to the information structure,

such that welfare satisfies

W = w(⇤0), (55)

with

⇤0 =

✓

⌃Q

+1

1� ↵�q

◆

+ ⇠

✓

⌃Y

+1

1� ↵�y

◆

, (56)

and where � and ✏ are given in (52) and ⇠ is a positive scalar pinned down by (�, ✏, ✓, ⌘).

Like Lemma 2 in the baseline model, Lemma 10 is not particularly surprising. It simply de-

composes the welfare losses that obtain relative to the first-best in two components. The first

16

component, namely the sum ⌃Q

+ 1

1�↵

�q

, capture the distortions (if any) that obtain in the first-

stage production decisions, that is, those that must be set on the basis of incomplete information.

The second component, namely the sum ⌃Y

+ 1

1�↵

�y

, captures the distortions (if any) that obtain

in the second-stage production decisions, that is, those that are free to adjust to the realized state.

Each of these components contains a volatility and a dispersion subcomponent, reflecting the fact

that some distortions are aggregate whereas others are idiosyncratic.

What is interesting, however, is how these components are a↵ected by the information frictions

and by the associated types of rigidity. To develop intuition, let us abstract from markup shocks.

When information is complete, all distortions vanish, ⌃Q

= ⌃q

= ⌃Y

= ⌃y

= 0, and hence ⇤0 = 0.

When, instead, information is incomplete, the nature of the distortions depends on whether the

incompleteness of information is only the source of real rigidity or also the source of nominal rigidity.

In the former case, ⌃Q

and ⌃q

are positive, reflecting the measurability constraint on quantities,

but ⌃Y

= ⌃y

= 0, reflecting the margin of adjustment in second-stage production to the realized

state. In the latter case, by contrast, whether ⌃Y

and ⌃y

coincide or diverge from zero depends

on whether monetary policy coincides or diverges from the benchmark of replicating flexible prices.

This discussion therefore underscores how the two types of rigidity map into di↵erent kinds of

potential distortions, an issue that is further explored in the main text.

B.3 An Example With Di↵erent Policy Targets

In the main text, we noted that, if monetary policy tries to stabilize either the price level or the

output gap, more precise information may help increase welfare not only by attenuating the real

rigidity but also by alleviating the policy suboptimality. We now illustrate the logic behind this

argument in Figure B1, with the help of a numerical example. This example assumes the following

parameterization: ✓ = ⌘ = 0.5, ✏ = 1, � = 0.25, ⇢ = 0.5, �a

= �x

= �z

= .02, and �m

= �✏

= 0.

The figure reports the welfare e↵ects of public information (namely the value of ⇤ against the

value of �z

) under three alternative specifications of the monetary policy: the optimal policy (solid

blue line), a policy that stabilizes the aggregate output gap (dotted red line), and a policy that

stabilizes the price level (dashed green line). The last two policies are suboptimal in our model

due to the presence of the informational friction, but are useful reference points because they are

optimal in the prototypical New-Keynesian model.

For any level of noise �z

> 0, the welfare losses associated with targeting either the price level

or the output gap are higher than those associated with the optimal policy. Nevertheless, the

welfare gap between these policies and the optimal one decreases with the precision of the available

information, and vanishes in the limit as �z

! 0. It follows that, under either of these two policies,

more information improves welfare via two e↵ects: by bringing the equilibrium allocation closer to

the optimal one; and by raising the welfare attained at the optimal allocation.

We hope that these findings give some guidance about the potential role of policies which are

17

0.01 0.02 0.03 0.04σz

0.02

0.04

0.06

0.08

Λ

������ ������ ��������� ����� ������������ ������

Figure 1: An illustration of the welfare losses due to incomplete information, under di↵erent levels

of noise in the public signal and under di↵erent policy rules.

not exactly optimal in our setting but are perhaps closer to real-world practice, such as a policies

that follow a Taylor rule. We nevertheless have to leave any serious quantitative exploration of the

issue for future work.

B.4 Proofs

In this subsection, we provide first the proofs of the auxiliary results we stated in parts B.1 and

B.2 of this appendix and next the proofs of the results that appear in Section 4.

Proof of Lemma 6. Part (i). The proof combines the first-order conditions of the firm’s problem

with the resource constraint and the monetary policy rule. These are used to derive the conditions

that the coe�cients in (43) have to satisfy in order to be part of an equilibrium.

We now seek to translate these properties in terms of the relevant coe�cients that parameterize

the allocations, prices, and policy under a log-normal specification. First note that, since all shocks

are independent over time, Et

[log Yt+1

] and Et

[logPt+1

] in (42) will be constant. Thus, let (omitting

unimportant constants)

log qit

= 's

sit

+ 'x

xit

+ 'z

zt

log lit

= ls

st

+ ls

sit

+ lx

xit

+ lz

zt

log Yt

= cs

st

+ cz

zt

log pit

= s

sit

+ x

xit

+ z

zt

for some coe�cients ('s

,'x

, ...., z

), and with the understanding that s stands either for a or µ,

depending on whether we are considering the case of technology or markup shocks. Note that the

18

resource constraint (40) is satisfied if and only if

cs

= ('s

+ 'x

) + ⌘(ls

+ lx

+ ls

) (57)

cz

= 'z

+ ⌘lz

(58)

while (42) is satisfied if and only if

⇢s

= ��cs

� ( s

+ x

) (59)

⇢z

= ��cz

� z

. (60)

Consider the consumer’s demand function (41), which can be expressed as follows:

�⇢ (log pit

� logPt

) = (log yit

� log Yt

)

Using market clearing, the production function, and the proposed strategies, we can express the

output of good i as follows:

log yit

= log qit

+ ⌘ log lit

= ('s

+ ⌘ls

)sit

+ ('x

+ ⌘lx

)xit

+ ('z

+ ⌘lz

)zt

+ ⌘ls

st

,

By implication, aggregate output satisfies

log Yit

= ('s

+ ⌘ls

+ 'x

+ ⌘lx

)st

+ ('z

+ ⌘lz

)zt

.

Substituting the above two results in the consumer’s demand function, and doing a similar sub-

stitution for pit

and Pt

, we infer that the following must hold for all realizations of shocks and

signals:

�⇢( s

sit

+ x

xit

� ( s

+ x

)st

) = ('s

+ ⌘ls

)sit

+ ('x

+ ⌘lx

)xit

� ('s

+ ⌘ls

+ 'x

+ ⌘lx

)st

.

This is true if and only if

s

= �1

⇢

('s

+ ⌘ls

) and x

= �1

⇢

('x

+ ⌘lx

). (61)

Finally, taking the logs of conditions (38) and (39) and using the properties of log-normal distribu-

tions, we can rewrite these conditions as follows:

Eit

[log lit

] = 1

✓

(log qit

� ait

) (62)

log qit

= �a

ait

+ �µ

µit

+ ↵

�

Eit

[log Yt

] . (63)

Clearly, condition (62) holds for all i if and only if

ls

= 1

✓

('s

� 1s=a

) (64)

lx

= 1

✓

'x

� ls

x

s

+

x

+

z

(65)

lz

= 1

✓

'z

� ls

z

s

+

x

+

z

, (66)

19

while condition (63) holds for all i if and only if

's

= �s

(67)

'x

= ↵

�

cs

x

s

+

x

+

z

(68)

'z

= ↵

�

⇣

cs

z

s

+

x

+

z

+ cz

⌘

. (69)

We can now use (57)-(58) to replace cs

and cz

in (68) and (69) to get

'x

= ↵

�

('s

+ 'x

+ ⌘ (ls

+ lx

+ ls

))

x

s

+

x

+

z

and

'z

= ↵

�

('s

+ 'x

+ ⌘ (ls

+ lx

+ ls

))

z

s

+

x

+

z

+ ↵

�

('z

+ ⌘lz

) ,

with the understanding that s stands for either a or µ, depending on the case under consideration.

Using (64), (65), (66), and (67) and rearranging gives⇣

s

+

x

+

z

x

�

↵

� 1� ⌘

✓

⌘

'x

= �s

+ ⌘

✓

⇣

�s

� 1s=a

⌘

+ ⌘

s

+

z

s

+

x

+

z

ls

and�

�

↵

� 1� ⌘

✓

�

'z

= �

↵

z

x

'x

� ⌘

z

s

+

x

+

z

ls

.

or

'x

= �x

+ �0x

ls

,

and

'z

= �z

+ �0z

ls

,

where

�x

⌘ˆ

�

s

+

⌘

✓

(ˆ�s

�1s=a

)

s

+

x

+

z

x

�

↵

�1�⌘

✓

, �0x

⌘ ⌘

x

s

+

x

+

z

s

+

z

(

s

+

x

+

z

)

�

↵

�

x

(1+ ⌘

✓

),

�z

⌘�

↵

z

x

�

↵

� 1� ⌘

✓

�x

, �0z

⌘�

↵

z

x

�0x

� ⌘

z

s

+

x

+

z

�

↵

� 1� ⌘

✓

.

This completes the proof of the necessity of conditions (44)-(49) for an allocation to be part of an

equilibrium.

We now prove su�ciency. Pick arbitrary ls

and let ('s

,'x

,'z

, ls

, lx

, lz

) be the unique vector that

satisfies conditions (44) through (49) for the given ls

. Next, let (cs

, cz

, ⇢s

, ⇢z

, s

, x

) be determined

as in (57)-(61) and let z

= ��cz

. By construction, the allocations, prices and policies defined in

this way constitute an equilibrium, which completes the su�ciency argument.

Part (ii). This proof is the same as that of part (i), except for one key di↵erence: now the

marginal costs and returns of second-state employment must be equated state-by-state, not just in

expectation. It is this additional restriction that pins down ls

at the value stated in equation (50)

in the lemma. A detailed derivation is available upon request. ⌅

20

Proof of Lemma 7. Before proceeding, it is useful to recall two key facts about Lemma 6. First,

Lemma 6 established that, when the interest rate is determined according to (42), an allocation

as in (43) can be implemented as part of an equilibrium if and only if conditions (44)-(49) are

satisfied. Second, the nominal prices that supported such an allocation (i.e., that were consistent

with the firm’s optimal price-setting behavior) were left outside the statement of that lemma, but

were constructed as part of its proof. With this in mind, the strategy underlying the present proof

is to show that the class of policies in (51) spans exactly the same set of allocations as the class of

policies in (42), but now the nominal prices that support any such allocation are di↵erent.

Thus let us start by collecting, once again, the key equilibrium conditions:

0 = n1+✏

it

� Eit

2

4Mit

U 0 (Yt

)

✓

yit

Yt

◆�1

⇢

✓cit

3

5 (70)

0 = Eit

2

4

0

@l1+✏

it

�Mit

U 0 (Yt

)

✓

yit

Yt

◆�1

⇢

⌘yit

1

A

3

5 (71)

Yt

=

Z

(qit

l⌘it

)⇢�1⇢ di

�

⇢

⇢�1

; (72)

pit

Pt

=

✓

qit

l⌘it

Yt

◆�1

⇢

; (73)

log(1 +Rt

) = � (Et

[log Yt+1

]� log Yt

) + (Et

[logPt+1

]� logPt

) (74)

Clearly, these conditions are necessary and su�cient for equilibrium regardless of how Rt

is deter-

mined.

Now, let us restrict Rt

to be determined according to (51) and let us consider the following

strategies (omitting unimportant constants):

log qit

= 's

sit

+ 'x

xit

+ 'z

zt

log lit

= ls

st

+ ls

sit

+ lx

xit

+ lz

zt

log pit

= s

sit

+ x

xit

+ z

zt

+ �k

st�k

for some coe�cients ('s

,'x

, ..., �k

). The proof proceeds by obtaining the restrictions that equi-

librium imposes on these coe�cients and by showing that the restrictions imposed on the '’s and

the l’s (the quantity-related coe�cients) are the same as those found in Lemma 6, whereas those

that are imposed on the ’s (the price-related coe�cients) are di↵erent. The proof then concludes

by obtaining the scalars ⇢�k

and ⇢z

that implement these coe�cients.

First, consider conditions (70) and (71), which encapsulated the optimal behavior of the firms

and the workers. These conditions can be restated as:

Eit

[log lit

] = 1

✓

(log qit

� ait

)

log qit

= �a

ait

+ �µ

µit

+ ↵

�

Eit

[log Yt

] .

21

Because the proposed strategies for qit

and lit

are the same as those in Lemma 6, and because

neither the prices nor the interest rate enter into the above conditions, these conditions reduce to

exactly the same restrictions on the � and the l coe�cients as those in Lemma 6.

Next, consider condition (72). Under the proposed strategies, this reduces to

log Yt

= cs

st

+ cz

zt

with

cs

= ('s

+ 'x

) + ⌘(ls

+ lx

+ ls

) and cz

= 'z

+ ⌘lz

(75)

Similarly to the case of the � and the l coe�cients, the restrictions that pertain to the coe�cients

cs

and cs

are therefore the same as the corresponding ones in the proof of Lemma 6.

Next, consider condition (73), which can be expressed as follows:

�⇢ (log pit

� logPt

) = (log yit

� log Yt

)

Under the proposed strategies, this reduces to the following:

�⇢( s

sit

+ x

xit

� ( s

+ x

)st

) = ('s

+ ⌘ls

)sit

+ ('x

+ ⌘lx

)xit

� ('s

+ ⌘ls

+ 'x

+ ⌘lx

)st

.

This in turn is true if and only if

s

= �1

⇢

('s

+ ⌘ls

) and x

= �1

⇢

('x

+ ⌘lx

). (76)

The restrictions on the coe�cients s

and x

are therefore also the same as the corresponding ones

in the proof of Lemma 6. (Also note that, up to this point, the coe�cients �k

and z

are “free”.)

Finally, consider the Euler condition (74). Let Xt

⌘ � log Yt

+logPt

and rewrite (74) as follows:

Xt

= � log(1 +Rt

) + Et

[Xt+1

].

Iterating this condition forward k times yields

Xt

= �k

X

j=0

Et

[log(1 +Rt+j

)] + Et

[Xt+k+1

] (77)

Under the proposed strategies, the output and price level in period t+ k + 1 satisfy

Et

[log Yt+k+1

]] = Et

[cs

st+k+1

+ cz

zt+k+1

] = constant

Et

[logPt+k+1

]] = Et

[( s

+ x

) st+k+1

+ z

zt+k+1

+ �k

st+1

]] = constant,

From (51), the interest rate satisfies

Et

[log(1 +Rt+j

)] = Et

[⇢�k

st�k+j

+ ⇢z

zt+j

]

=

8

>

>

<

>

>

:

⇢�k

st�k

+ ⇢z

zt

,

constant,

constant+ ⇢�k

st

,

for j = 0

for j 2 {1, ..., k � 1}for j = k

22

It follows that (77) reduces to the following (omitting constants):

Xt

= �⇢�k

st�k

� ⇢z

zt

� ⇢�k

st

At the same time, the proposed strategies imply

Xt

⌘ � log Yt

+ logPt

= � (cs

st

+ cz

zt

) + (( s

+ x

) st

+ z

zt

+ �k

st�k

)

= (�cs

+ s

+ x

) st

+ (�cz

+ z

) zt

+ �k

st�k

The two are consistent if and only if the following hold true:

�k

= �cs

+ s

+ x

(78)

⇢�k

= � �k

(79)

⇢z

= ��cz

� z

(80)

Now, pick any allocation that is implementable under (42). This is equivalent to choosing an

arbitrary value for ls

and letting the � and l coe�cients be determined by conditions (44)-(49). Any

such choice also pins down the values of cs

, cz

, s

, and x

according to the restrictions (75)-(76).

But then this pins down �k

from (78), which in turn pins down ⇢�k

from (79). We have thus

arrived to the value of ⇢�k

which is necessary and su�cient for the policy in (51) to implement the

allocation under consideration. Finally, what we are left with then is the familiar indeterminacy of

the response of the price level to the public signal: for any allocation that has been implemented,

we can still pick an arbitrary z

and let ⇢z

be determined from (80). ⌅

Proof of Lemma 8. Since prices are set without any information about the monetary shock (a

maintained assumption throughout), employment and aggregate output will respond to rt

one-to-

one. Thus, if we conjecture

log lit

= ls

st

+ ls

sit

+ lx

xit

+ lz

zt

+ lr

rt

log Yt

= cs

st

+ cz

zt

+ cr

rt

and follow steps analogous to those in the proof of part (i) of Lemma 6, we obtain cr

= ⌘lm

= �1/�.

Modulo these changes, it is immediate to see that conditions (44)-(49) remain true. ⌅

Proof of Lemma 9. As in the baseline model, once we use the equilibrium conditions for the

wages and the prices, the first-best level of output conditional on log qit

satisfies the following FOC

of the firm’s profit with respect to the second input:

l✏it

= U 0 (Yt

)

✓

yit

Yt

◆�1

⇢

✓

⌘yit

lit

◆

. (81)

23

Furthermore, the first-best level of output satisfies (81) and, in addition, the following FOC of the

firm’s profit with respect to the first input:

n✏

it

= U 0 (Yt

)

✓

yit

Yt

◆�1

⇢

✓

✓yit

nit

◆

(82)

Note that, by definition of the first-best level of output, the markup and the expectation operator

are absent from both conditions.

Rearranging (81) to solve for log lit

(and omitting unimportant constants)

(1 + ✏) log lit

=⇣

1

⇢

� �⌘

log Yt

+⇣

1� 1

⇢

⌘

log yit

(83)

and using log yit

= log qit

+ ⌘ log lit

to replace lit

, we can restate the above as

log yit

= ⇢⌫ log qit

+ ↵ log Yt

,

which proves (53). Using (53) to replace yit

in (82), taking logs, and noting that log qit

= ait

+

✓ log nit

, we arrive at

log q⇤it

= �a

ait

+ ↵

�

log Yt

, (84)

where �a

, ↵, �, and ⌫ are defined in (52). Finally, integrating (83) across di↵erent islandsZ

log lit

di =1� �

1 + ✏log Y

t

and combining the latter with (40) gives

log Yt

= � logQt

and therefore (84) gives (54). ⌅

Proof of Lemma 10. As with the proof of Lemma 2, we drop the time index t and focus on the

characterization of the per-period welfare flow. The latter is now defined as

W = E"

Y 1��

1� �� 1

1 + ✏

Z

✓

qi

Ai

◆

1+✏

✓

di� 1

1 + ✏

Z

✓

yi

qi

◆

1+✏

⌘

di

#

We will proceed to establish that the above can be expressed as follows:

W = W ⇤ exp

✓

�1

2(1� �) (1 + ✏)⇤0

◆

.

where W ⇤ is the first-best level of W .

We first focus on the first and third component of W :

E"

Y 1��

1� �� 1

1 + ✏

Z

✓

yi

qi

◆

1+✏

⌘

di

#

.

24

This expression resembles the expression for welfare in the baseline model with qi

replacing local

productivity ai

. We can thus follow the same steps as in the proof of Lemma 2 and rewrite the

latter expression as (omitting unimportant constants)

E [Q]

1��

1+✏

⌘

�1+�

1+✏

1�⌘

exp

�1

2

1+✏

⌘

(1��)

1+✏

⌘

�1+�

⌦

!

exp⇣

�1

2

1+✏

⌘

(1� �)⇣

⌃Y

+ 1

1�↵

�y

⌘⌘

,

with

⌦ ⌘⇣

1+✏

⌘

� + � � 1⌘

1+✏

⌘

+��1

⇣1+✏

⌘

⌘2 V ar�

log Y�

=1+✏

⌘

�+��1

1+✏

⌘

+��1

V ar (logQ) ,

where the second line uses the fact that (53) implies log Y = ⇢⌫

1�↵

logQ.

We can use these results to rewrite W as follows:

W = E [Q]1�� exp

�1

2

1+✏

⌘

(1��)

1+✏

⌘

�1+�

⌦� 1

2

1+✏

⌘

(1� �)⇣

⌃Y

+ 1

1�↵

�y

⌘

!

� 1

1+✏

E"

Z

⇣

q

i

A

i

⌘

1+✏

✓

di

#

Using property (15),

E [Q]1�� = Eh

Q1��

i

e1

2

�(1��)V ar(logQ)

we have that

W = 1

1��

Eh

Q1��

i

⌅� 1

1+✏

Z

⇣

q

i

A

i

⌘

1+✏

di

where

⌅ ⌘ (1� �) ✓ exp

1

2

� (1� �)V ar (logQ)� 1

2

1+✏

⌘

(1��)

1+✏

⌘

�1+�

⌦� 1

2

1+✏

⌘

(1� �)⇣

⌃Y

+ 1

1�↵

�y

⌘

!

,

or, using the definition of ⌦,

⌅ = (1� �) ✓ exp

✓

�1

2

1 + ✏

⌘(1� �)

✓

⌃Y

+1

1� ↵�y

◆◆

.

Note then that the functional that maps the strategy q into the welfare level W is the same as the

one in the proof of Lemma 2, provided that we make two changes: we replace ⇢, �, and ✏ with,

respectively, ⇢, �, and ✏; and we accommodate the constant ⌅. Thus, following the same steps as

in that proof, we can obtain the following characterization of the per-period welfare flow:

W = W ⇤ exp

⇢

�1

2

(1 + ✏) (1� �)

✓

⌃Q

� 1

1� ↵�q

◆�

⌅1+✏

✏+� .

Finally, using the definition of ⌅ and rearranging gives us

W = W ⇤ expn

�1

2

(1� �) (1 + ✏)⇣

⌃Q

� 1

1�↵

�q

⌘o

exp⇣

�1

2

1+✏

✏+�

(1� �) 1+✏

⌘

⇣

⌃Y

+ 1

1�↵

�y

⌘⌘

= W ⇤ expn

�1

2

(1� �) (1 + ✏)h⇣

⌃Q

� 1

1�↵

�q

⌘

+ ⇠⇣

⌃Y

+ 1

1�↵

�y

⌘io

,

where

⇠ ⌘1+✏

⌘

+ � � 1

✏+ �.

The result then follows from translating the above in terms of life-time welfare and defining the

function w (·) as w(x) ⌘ W⇤ exp�

�1

2

(1� �) (1 + ✏)x�

. ⌅

25

Proof of Lemma 3. Part (i). This part follows trivially from projecting (regressing) the equilib-

rium nominal GDP on the fundamental and the public signal, and letting (�s

,�z

) be the projection

coe�cients and mt

the residual.

Part (ii). From Lemma 8, the sum cs

+ ( s

+ x

), which gives the response of logMt

to the

fundamental, is a linear function of ⇢s

. Furthermore, cz

is invariant to ⇢z

, whereas z

is inversely

related to it. It follows that, for any pair (�s

,�z

), we can find a value of the pair (⇢s

, ⇢z

) so that

cs

+ ( s

+ x

) = �s

and cz

+ z

= �z

The result then follows from considering the policy that is identified by the combination of this

particular value for the pair (⇢s

, ⇢z

) with rt

= � 1

�

mt

.

Part (iii). This follows directly from the analysis in Section B.1. ⌅

Proof of Lemma 4. From part (ii) of Lemma 6 and the proof of Lemma 3 we conclude that

a monetary policy as in (7) replicates flexible-price allocations if and only if �s

= �⇤s

, where

�⇤s

⌘ ⇧+⇧l⇤s

and l⇤s

is given by (50). We can then use Lemma 3 to translate these results in terms

of the interest rate rule. ⌅

Proof of Proposition 5. Part (i). As in the proof of Lemma 6, once we use the equilibrium

conditions for the wages and the prices, the FOCs of the firm’s profit with respect to the two inputs

reduce to the following:

n✏

it

= Eit

2

4Mit

U 0 (Yt

)

✓

yit

Yt

◆�1

⇢

✓

✓yit

nit

◆

3

5 (85)

l✏it

= Mit

U 0 (Yt

)

✓

yit

Yt

◆�1

⇢

✓

⌘yit

lit

◆

(86)

Note that the first condition is the same as (38). The second condition, by contrast, does not

feature an expectation operator, because the absence of nominal rigidity means that the stage-2

input adjusts so as to equate marginal cost and marginal revenue state-by-state. Rearranging (86)

to solve for log lit

(and omitting unimportant constants)

(1 + ✏) log lit

= �µit

+⇣

1

⇢

� �⌘

log Yt

+⇣

1� 1

⇢

⌘

log yit

(87)

and using log yit

= log qit

+ ⌘ log lit

to replace lit

, we can restate the above as

log yit

= bq

log qit

� bµ

µit

+ ↵ log Yt

where

bq

⌘ ⇢⌫ and bµ

⌘ ⌘

1+✏

⇢⌫.

26

Using the above result to replace yit

in (85), taking logs, and noting that log qit

= ait

+ ✓ log nit

,

we arrive at

log qit

= �a

ait

+ �µ

µit

+ ↵

�

Eit

[log Yt

],

where �a

, �µ

, ↵, �, and ⌫ are defined in (52). Finally, integrating (87) across di↵erent islandsZ

log lit

di =1� �

1 + ✏log Y

t

� 1

1 + ✏µt

and combining the latter with (40) gives us

log Yt

= �⇣

logQt

� ⌘

1+✏

µt

⌘

.

Thus,

log qit

= �a

ait

+ �µ

µit

+ �µ

Eit

[µt

] + ↵Eit

[logQt

] ,

where �µ

⌘ �↵ ⌘

1+✏

.

Part (ii). From Lemma 10 we have that, irrespective of whether the nominal rigidity is present

or not, welfare is a decreasing function of

⇤0 =

✓

⌃Q

+1

1� ↵�q

◆

+ ⇠

✓

⌃Y

+1

1� ↵�y

◆

.

When the policy of Lemma 4 is in place, the equilibrium quantities satisfy the FOC (86) which, as

shown in part (i), can be rearranged as

log yit

= bq

log qit

� bµ

µit

+ ↵ log Yt

.

On the other hand, Lemma 9 shows that the first-best level of output conditional on the first-period

equilibrium production decision satisfies

log yit

= bq

log qit

+ ↵ log Yt

.

Therefore, the aggregate and local gaps are given by, respectively,

log Yt

� log Yt

= �bµ

µt

where bµ

⌘ b

µ

1�↵

and

log yit

� log yit

= �bµ

µit

� ↵bµ

µt

.

Importantly, note that bµ

and bµ

are independent of the information structure. It follows that the

second term in ⇤0 reduces to

⌃Y

+1

1� ↵�y

= b2µ

�2µ

+1

1� ↵b2µ

�2⇠

which is independent of the information structure. Therefore, (56) reduces to

⇤0 = ⌃Q

+1

1� ↵�q

+ !

where ! ⌘ ⇠b

2µ

1�↵

⇣

1

1�↵

�2µ

+ �2⇠

⌘

, corresponding to equation (11) in the main text. ⌅

27

Proof of Theorem 3. From Proposition 5, when the policy that replicates flexible prices is in

place, welfare is a decreasing function of

⇤0 = ⌃+1

1� ↵� + !.

Since ! is independent of the information structure, the e↵ects of the precision of private and public

information are determined only by the changes in the volatility and dispersion of the first-period

gaps.

Consider first the case with technology shocks. In that case �µ

= �µ

= 0 and ! = 0 and the

welfare e↵ects coincide with those found in Theorem 1.

In the case of markup shocks, �µ

is given in (52), �µ

⌘ �↵ ⌘

1+✏

, ! is given in Proposition 5, and

⇤0 =(↵⌘�(1+✏)

ˆ

�

µ

)2((1�↵)

x

+

z

)�(1�↵)(1+✏)(2↵⌘�(1+✏)(1+↵)

ˆ

�

µ

)µ

ˆ

�

µ

(1�↵)

2(1+✏)

2(

µ

+(1�↵)

x

+

z

)

µ

+ˆ

�

2µ

(1�↵)

⇠

+ !.

Di↵erentiating ⇤0 with respect to the precision of, respectively, public and private information gives

@⇤0

@z

=↵

2(⌘�(1+✏)

ˆ

�

µ

)2

(1�↵)

2(1+✏)

2(

µ

+(1�↵)

x

+

z

)

2

and@⇤0

@x

=↵

2(⌘�(1+✏)

ˆ

�

µ

)2

(1�↵)(1+✏)

2(

µ

+(1�↵)

x

+

z

)

2 ,

which are both always positive. ⌅

Proof of Proposition 6. While derivations are lengthy, the idea of the proof is simple. From

Lemma 10 we know that welfare losses ⇤0 depend on the volatility and dispersion of the first-period

and second-period gaps between equilibrium and first-best production. We thus take each gap and

decompose it into two new gaps: the first captures the deviation of equilibrium production from

production with flexible prices; and the second captures the deviation of flexible-price production

from first-best production. Finally, we rewrite the volatility and dispersion of the original gaps in

terms of the volatility and dispersion of the new gaps.

We introduce some notation which will simplify the expressions in the proof. First, we let qit

denote first-period output when monetary policy replicates flexible prices. Next, we decompose the

first-period output gap as follows

log qit

⌘ log qit

� log q⇤it

= log qit

+ log qit

,

where log qit

⌘ log qit

� log qit

denotes the deviation of first-period equilibrium production from

the flexible-price benchmark, and where log qit

⌘ log qit

� log q⇤it

denotes the usual output gap with

flexible prices. With the same notation we obtain a similar decomposition for the second-period

output gap:

log yit

⌘ log yit

� log yit

= log yit

+ log yit

.

28

Finally, in the proof we use analogous decompositions for the first and second-period aggregate

gaps, log Qt

and log Yt

.

The volatility of the first-period aggregate gap can thus be expressed as follows

⌃Q

⌘ V ar⇣

log Qt

⌘

= V ar⇣

log Qt

⌘

+ 2Cov⇣

log Qt

, log Qt

⌘

+ ⌃Q

,

where ⌃Q

is the aggregate volatility of the first-period aggregate gap with flexible prices. Similarly,

the dispersion of the first-period local gaps can be rewritten as

�q

⌘ V ar⇣

log qit

� log Qt

⌘

= V ar⇣

log qit

� log Qt

⌘

+ 2Cov⇣

log qit

� log Qt

, log qit

� log Qt

⌘

+ �q

,

where �q

is the dispersion of the first-period local gaps with flexible prices.

From part (i) of Lemma 6, first-period production satisfies (43) with 'x

and 'z

given by (45) and

(46), respectively. We denote by 'x

and 'z

the coe�cients 'x

and 'z

when the policy replicating

flexible prices is in place. Combining parts (i) and (iii) of Lemma 6, 'x

and 'z

satisfy

'x

= �x

+ �0x

l⇤a

=ˆ

�

a

a

+(1�↵)

x

+

z

↵x

,

'z

= �z

+ �0z

l⇤a

=ˆ

�

a

a

+(1�↵)

x

+

z

↵

z

1�↵

.

Finally, first-best production is given by (54). We can thus compute all the gaps defined above as

follows

log Qt

= Qa

a

at

+ Qa

z

"t

log Qt

= Qa

a

at

+ Qa

z

"t

,

where Qa

a

⌘ �0x

+ �0z

, Qa

z

⌘ �0z

, Qa

a

⌘ � ↵

1�↵

�a

+ �x

+ �0x

l⇤a

+ �z

+ �0z

l⇤a

, and Qa

z

⌘ �z

+ �0z

l⇤a

.

Therefore,

V ar⇣

log Qt

⌘

=⇣

Qa

a

⌘

2

(la

� l⇤a

)2 1

a

+⇣

Qa

z

⌘

2

(la

� l⇤a

)2 1

z

Cov⇣

log Qt

, log Qt

⌘

= Qa

a

Qa

a

(la

� l⇤a

)1

a

+ Qa

z

Qa

z

(la

� l⇤a

) 1

z

.

Similarly, the terms capturing the dispersion of the first-period local gaps can be expressed as

log qit

� log Qt

= qax

(la

� l⇤a

)uit

log qit

� log Qt

= qax

uit

.

where qax

⌘ �0x

and qax

⌘ �x

+ �0x

l⇤a

. Therefore,

V ar⇣

log qit

� log Qt

⌘

= (qax

)2 (la

� l⇤a

)21

x

Cov⇣

log qit

� log Qt

, log qit

� log Qt

⌘

= qax

qax

(la

� l⇤a

) 1

x

.

29

Furthermore, it turns out that, in the case of technology shocks, the covariance terms do not

contribute to the welfare losses, that is,

Cov⇣

log Qt

, log Qt

⌘

+1

1� ↵Cov

⇣

log qit

� log Qt

, log qit

� log Qt

⌘

=

✓

Qa

a

Qa

a

1

a

+ Qa

z

Qa

z

1

z

+1

1� ↵qax

qax

1

x

◆

(la

� l⇤a

) = 0.

Collecting all the remaining terms together, the second-order welfare losses associated with the

first-period gaps can be rewritten as

⌃Q

+1

1� ↵�q

= #a

(la

� l⇤a

)2 + ⌃Q

+1

1� ↵�q

, (88)

where

#a

⌘⇣

Qa

a

⌘

2

1

a

+⇣

Qa

z

⌘

2

1

z

+ 1

1�↵

(qax

)2 1

x

.

Let’s now turn to the second-period output gaps. With technology shocks flexible-price alloca-

tions satisfy (86) with a constant markup. If we rearrange (86) we can then obtain an expression

similar to (53), except for a constant capturing the markup. Thus, with technology shocks flexible-

price allocations and first-best allocations conditional on equilibrium first-period production di↵er

only by a constant and, thus, there is no use in decomposing the second-period gaps as we did for

the first-period gaps. Using the conditions in Lemma 6, the second-period aggregate and local gaps

are, respectively,

log Yt

= Y a

a

(la

� l⇤a

) at

+ Y a

z

(la

� l⇤a

) "t

log yit

= Y a

x

(la

� l⇤a

)uit

where Y a

a

⌘�

1 + ⌘

✓

�

�0x

+�

1 + ⌘

✓

�

�0z

+ ⌘

a

a

+

x

+

z

, Y a

z

⌘�

1 + ⌘

✓

�

�0z

� ⌘

z

a

+

x

+

z

, and Y a

x

⌘�

1 + ⌘

✓

�

�0x

�⌘

x

a

+

x

+

z

. Thus, volatility and dispersion of second-period gaps are, respectively,

⌃Y

=⇣

Y a

a

⌘

2

(la

� l⇤a

)2 1

a

+⇣

Y a

z

⌘

2

(la

� l⇤a

)2 1

z

+ �2m

�y

=⇣

Y a

x

⌘

2

(la

� l⇤a

)2 1

x

.

Collecting all terms, the second-order welfare losses associated with the second-period gaps can be

rewritten as

⌃Y

+1

1� ↵�y

= #0a

(la

� l⇤a

)2 + �2m

,

where

#0a

⌘⇣

Y a

a

⌘

2

1

a

+⇣

Y a

z

⌘

2

1

z

+ 1

1�↵

⇣

Y a

x

⌘

2

1

x

.

Finally, in the proof of Lemma 8 we show that we can rewrite la

� l⇤a

as (�s

� �⇤s

) /⇧. Therefore,

if we let ⇤ = ⌃Q

+ 1

1�↵

�q

, T = �2m

, and

⇥ ⌘ #a

+ ⇠#0a

⇧2�2a

,

30

the result, for the case with technology shocks, follows directly from the welfare decomposition in

Lemma 10.

The proof for the case with markup shocks closely resembles the proof with technology shocks;

here we simply report the terms required to derive K and T . Since markups are absent from

first-best allocations, the latter are constant when the business cycle is driven by markup shocks.

Using parts (i) and (iii) of Lemma 6, the first-period aggregate gap log Qt

is given by the sum of

the following gaps

log Qt

= Qµ

µ

�

lµ

� l⇤µ

�

µt

+ Qµ

z

�

lµ

� l⇤µ

�

"t

log Qt

= Qµ

µ

µt

+ Qµ

z

"t

.

where Qµ

µ

⌘ �0x

+ �0z

, Qµ

z

⌘ �0z

, Qµ

µ

⌘ �µ

+ �x

+ �0x

l⇤µ

+ �z

+ �0z

l⇤µ

, and Qµ

z

⌘ �z

+ �0z

l⇤µ

. Therefore,

V ar�

log Qt

�

=⇣

Qµ

µ

⌘

2

1

µ

�

lµ

� l⇤µ

�

2

+⇣

Qµ

z

⌘

2

1

z

�

lµ

� l⇤µ

�

2

Cov⇣

log Qt

, log Qt

⌘

= Qµ

µ

Qµ

µ

�

lµ

� l⇤µ

�

1

µ

+ Qµ

z

Qµ

z

�

lµ

� l⇤µ

�

1

z

.

Similarly, the terms capturing the dispersion of the first-period local gaps can be expressed as

log qit

� log Qt

= qµx

�

lµ

� l⇤µ

�

uit

log qit

� log Qt

= qµ⇠

⇠it

+ qµx

uit

,

where qµx

⌘ �0x

, qµ⇠

⌘ �µ

, and qµx

⌘ �x

+ �0x

l⇤µ

. Therefore,

V ar⇣

log qit

� log Qt

⌘

= (qµx

)2 1

x

�

lµ

� l⇤µ

�

2

Cov⇣

log qit

� log Qt

, log qit

� log Qt

⌘

= qµx

⇣

qµ⇠

+ qµx

⌘

1

x

�

lµ

� l⇤µ

�

.

Collecting all terms, the second-order welfare losses associated with the first-period gaps can be

rewritten as

⌃Q

+1

1� ↵�q

= #µ

�

lµ

� l⇤µ

�

2

+ 2#µ

�

lµ

� l⇤µ

�

+ ⌃Q

+1

1� ↵�q

,

where

#µ

⌘⇣

Qµ

µ

⌘

2

1

µ

+⇣

Qµ

z

⌘

2 1

z

+1

1� ↵(qµ

x

)2 1

x

#µ

⌘ Qµ

µ

Qµ

µ

1

µ

+ Qµ

z

Qµ

z

1

z

+1

1� ↵qµx

⇣

qµ⇠

+ qµx

⌘

1

x

.

Let us now turn to the second-period output gaps. With markup shocks, flexible-price output

and first-best output conditional on first-period equilibrium production are no longer related only

by a constant. In particular, the latter satisfies (53) and the corresponding aggregate level is

obtained by aggregating (53) across islands:

log Yt

=⇢⌫

1� ↵logQ

t

.

31

In contrast, equilibrium and flexible-price allocations can be obtained using the conditions in parts

(i), (iii), and (iv) of Lemma 6.

Combining terms, the second-period aggregate gaps are given by the following:

log Yt

= Y µ

µ

�

lµ

� l⇤µ

�

µt

+ Y µ

z

�

lµ

� l⇤µ

�

"t

+mt

log Yt

= Y µ

x

µt

.

where Y µ

µ

⌘�

1 + ⌘

✓

�

(�0x

+ �0z

) + ⌘

µ

µ

+

x

+

z

, Y µ

z

⌘�

1 + ⌘

✓

�

�0z

� ⌘

z

µ

+

x

+

z

, and Y µ

x

⌘ � ⌘

1+"

⇢⌫

1�↵

.

Therefore,

V ar⇣

log Yt

⌘

=⇣

Y µ

µ

⌘

2

1

µ

�

lµ

� l⇤µ

�

2

+⇣

Y µ

z

⌘

2

1

z

�

lµ

� l⇤µ

�

2

+ �2m

Cov⇣

log Yt

, log Yt

⌘

= Y µ

x

Y µ

µ

1

µ

�

lµ

� l⇤µ

�

.

Similarly, the terms capturing the dispersion of the second-period local gaps can be expressed as

log yit

� log Yt

= yµx

�

lµ

� l⇤µ

�

uit

log yit

� log Yt

= yµ⇠

⇠it

,

where yµx

⌘�

1 + ⌘

✓

�

�0x

� ⌘

x

µ

+

x

+

z

and yµx

⌘ � ⌘

1+"

⇢⌫. Therefore,

V ar⇣

log yit

� log Yt

⌘

= (yµx

)2 1

x

�

lµ

� l⇤µ

�

2

Cov⇣

log yit

� log Yt

, log yit

� log Yt

⌘

= yµ⇠

yµx

1

x

�

lµ

� l⇤µ

�

.

Collecting all terms together, the second-order welfare losses associated with the second-period

gaps can be rewritten as

⌃Y

+1

1� ↵�y

= #0µ

�

lµ

� l⇤µ

�

2

+ 2#0µ

�

lµ

� l⇤µ

�

+ ⌃Y

+1

1� ↵�y

+ �2m

,

where

#0µ

⌘⇣

Y µ

µ

⌘

2

1

µ

+⇣

Y µ

z

⌘

2

1

z

+1

1� ↵(yµ

x

)2 1

x

#0µ

⌘ Y µ

x

Y µ

µ

1

µ

+1

1� ↵yµ⇠

yµx

1

x

.

Finally, in the proof of Lemma 8 we show that we can rewrite lµ

� l⇤µ

as (�s

� �⇤s

) /⇧, thus, if we

let

⇤ = ⌃Q

+1

1� ↵�q

+ ⇠

✓

⌃Y

+1

1� ↵�y

◆

, T = �2m

,

⇥1

=#µ

+ ⇠#0µ

⇧2�2µ

, and ⇥2

= �#µ

+ ⇠#0µ

⇧2�2µ

,

the statement of the proposition follows directly from the welfare decomposition in Lemma 10. ⌅

32

Proof of Lemma 5. From Proposition 6, the optimal policy can be found by minimizing the

term ⇤+K+ T . The only terms that depend on the monetary policy are K and T . The minimum

value for T is clearly achieved when �2m

= 0. From (12), �⇤⇤s

= argmin�

s

K (�s

) = ⇥1

/⇥2

and the

minimum value is K (x

,z

) = �⇥2

1

/⇥2

. Finally, note that, since K (0) = 0, it has to be the case

that K (x

,z

) 0.

Part (i). When real rigidities are absent (✓ = 0),

K (x

,z

) = � (1� �⇢)2

(� + ✏)2 (1 + ✏⇢) ((1 + ✏⇢) (µ

+ z

) + ⇢ (� + ✏)x

),

which is clearly increasing in both x

and z

.

Part (ii). This is proved by example. When

(�, ✏, ⌘, ✓, ⇢,s

,⇠

,x

,z

) = (1, 1, .5, .5, 2, 0, .1, .1, .9) ,

K (x

,z

) is strictly increasing in both x

and z

. On the contrary, when

(�, ✏, ⌘, ✓, ⇢,s

,⇠

,x

,z

) = (2, 1, .5, .5, 2, 0, .5, .5, .5) ,

K (x

,z

) is striclty decreasing in x

; and when

(�, ✏, ⌘, ✓, ⇢,s

,⇠

,x

,z

) = (.1, .45, .03, .36, .09, 0, .1, .1, .9) ,

K (x

,z

) is strictly decreasing in z

. (By continuity, the aforementioned monotonicities continue

to hold in the neighborhood of the considered parameter values.)

Part (iii). First note that both ⇥1

and ⇥2

are linear in = x

+ z

. Thus, K(, %) is linear in

and the statement follows from the fact that �⇥2

1

/⇥2

0. ⌅

Proof of Theorem 4. Part (i). This follows from Proposition 5 along with the fact that, in the

case of technology shocks, the optimal policy replicates flexible prices.

Part (ii). This follows from Proposition 5, Lemma 5, and the discussion in the main text. ⌅

33