Online Appendix to Exogenous Information, Endogenous Information and Optimal Monetary Policy Luigi Paciello

Mirko Wiederholt

Einaudi Institute for Economics and Finance

Goethe University Frankfurt

October 2012

Abstract This document contains the online Appendices A-I for the October 2012 version of the paper “Exogenous Information, Endogenous Information and Optimal Monetary Policy.”

1

A

The central bank’s objective

Proposition 1 (The central bank’s objective) Let u denote the period utility function given by equation (19) in the paper. Let u ˜ denote the second-order Taylor approximation to this function u at the origin. Let E denote the unconditional expectation operator. Let xt , zt , and ω t denote the following vectors ³

x0t = zt0 ω 0t

ct cˆ1,t · · · cˆI−1,t ´ = at λt , ³ ´ = x0t zt0 1 . ³

´

,

Let ωn,t denote the nth element of ωt . Suppose there exist two constants δ < (1/β) and φ ∈ R such that, for all n and k and each period t ≥ 0, E |ω n,t ωk,t | < δ t φ. Then E

"∞ X t=0

t

#

βu ˜ (xt , zt ) − E

"∞ X

β

t

#

u ˜ (x∗t , zt )

t=0

¸ 1 ∗ 0 ∗ (xt − xt ) H (xt − xt ) , = βE 2 t=0

where the matrix H is given by ⎡ 1 0 ⎢ γ − 1 + α (1 + ψ) ⎢ 1+Λ−α ⎢ 0 2 I(1+Λ)α ⎢ ⎢ .. 1+Λ−α H = −C 1−γ ⎢ . ⎢ I(1+Λ)α ⎢ .. .. ⎢ . . ⎢ ⎣ 1+Λ−α 0 I(1+Λ)α

and the vector x∗t is given by

c∗t = and

(1)

∞ X

∙

t

···

···

1+Λ−α I(1+Λ)α

..

.

..

.

··· .. . .. . 1+Λ−α I(1+Λ)α

1+Λ−α 2 I(1+Λ)α

1+Λ−α I(1+Λ)α

...

0 .. .

1+Λ−α I(1+Λ)α

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

1 α

(1 + ψ) at , γ − 1 + α1 (1 + ψ) cˆ∗i,t = 0.

Proof. First, we introduce notation. The vector xt denotes the vector of all arguments of the function u that are endogenous variables ³ xt = ct cˆ1,t · · · 2

cˆI−1,t

´0

.

The vector zt denotes the vector of all arguments of the function u that are exogenous variables zt =

³

at λt

´0

.

Second, we compute a log-quadratic approximation to the period utility function (18) in the paper around the non-stochastic steady state. Recall that u ˜ denotes the second-order Taylor approximation to the function u at the origin. We have # "∞ X t βu ˜ (xt , zt ) E t=0

"∞ X

µ ¶# 1 0 1 0 0 0 0 = E β u (0, 0) + hx xt + hz zt + xt Hx xt + xt Hxz zt + zt Hz zt , 2 2 t=0 t

(2)

where hx denotes the vector of first derivatives of u with respect to xt evaluated at the origin, hz is the vector of first derivatives of u with respect to zt evaluated at the origin, Hx is the matrix of second derivatives of u with respect to xt evaluated at the origin, Hz is the matrix of second derivatives of u with respect to zt evaluated at the origin, and Hxz is the matrix of second derivatives of u with respect to xt and zt evaluated at the origin. Third, we rewrite equation (2) using condition (1). Condition (1) implies that ¯ ¯ ∞ X ¯ ¯ 1 0 1 0 t ¯ 0 0 0 β E ¯u (0, 0) + hx xt + hz zt + xt Hx xt + xt Hxz zt + zt Hz zt ¯¯ < ∞. 2 2 t=0

It follows that one can change the order of integration and summation on the right-hand side of equation (2): E

"∞ X t=0

=

∞ X t=0

t

#

βu ˜ (xt , zt )

∙ ¸ 1 0 1 0 0 0 0 β E u (0, 0) + hx xt + hz zt + xt Hx xt + xt Hxz zt + zt Hz zt . 2 2 t

(3)

See Rao (1973), p. 111. Condition (1) also implies that the infinite sum on the right-hand side of equation (3) converges to an element in R. Fourth, we define the vector x∗t . In each period t ≥ 0, the vector x∗t is defined by hx + Hx x∗t + Hxz zt = 0.

(4)

We will show below that Hx is an invertible matrix. Therefore, one can write the last equation as x∗t = −Hx−1 hx − Hx−1 Hxz zt . 3

Hence, x∗t is uniquely determined and the vector ω t with xt = x∗t satisfies condition (1). Fifth, equation (3) implies that # "∞ # "∞ X X t t ∗ βu ˜ (xt , zt ) − E βu ˜ (xt , zt ) E t=0

t=0

¸ ∙ 1 0 1 ∗0 t 0 ∗ ∗ ∗ 0 = β E hx (xt − xt ) + xt Hx xt − xt Hx xt + (xt − xt ) Hxz zt . 2 2 t=0 ∞ X

Using equation (4) to substitute for Hxz zt in the last equation and rearranging yields "∞ # "∞ # X X E βtu ˜ (xt , zt ) − E βtu ˜ (x∗t , zt ) t=0

=

∞ X t=0

t=0

¸ 1 ∗ 0 ∗ βE (xt − xt ) Hx (xt − xt ) . 2 t

∙

(5)

Sixth, we compute the vector of first derivatives and the matrices of second derivatives appearing in equations (4) and (5). We obtain hx = 0,

⎡

and

1 0 ⎢ γ − 1 + α (1 + ψ) ⎢ 1+Λ−α ⎢ 0 2 I(1+Λ)α ⎢ ⎢ .. 1+Λ−α Hx = −C 1−γ ⎢ . ⎢ I(1+Λ)α ⎢ .. .. ⎢ . . ⎢ ⎣ 1+Λ−α 0 I(1+Λ)α

⎡

Hxz

⎢ ⎢ ⎢ ⎢ ⎢ 1−γ ⎢ =C ⎢ ⎢ ⎢ ⎢ ⎣

1 α

···

···

..

.

..

.

··· .. . .. .

1+Λ−α I(1+Λ)α

1+Λ−α I(1+Λ)α

1+Λ−α 2 I(1+Λ)α

1+Λ−α I(1+Λ)α

... ⎤

(1 + ψ) 0 ⎥ ⎥ 0 0 ⎥ ⎥ .. .. ⎥ . . ⎥ ⎥. .. .. ⎥ ⎥ . . ⎥ ⎦ 0 0

0 .. .

1+Λ−α I(1+Λ)α

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

(6)

(7)

(8)

Seventh, substituting equations (6)-(8) into equation (4) yields the following system of I equations: c∗t =

1 α

(1 + ψ) at , γ − 1 + α1 (1 + ψ)

and, for all i = 1, . . . , I − 1, cˆ∗i,t +

I−1 X k=1

4

cˆ∗k,t = 0.

(9)

(10)

Finally, we rewrite equation (10). Summing equation (10) over all i 6= I yields I−1 X

cˆ∗i,t = 0.

i=1

Substituting the last equation back into equation (10) yields cˆ∗i,t = 0. Collecting equations (5), (7), (9), and (11), we arrive at Proposition 1.

5

(11)

B

Proof of Proposition 5

Step 1: Equilibrium for a given policy. This step is identical to Step 1 in the proof of Proposition 4. The reason is simple. The only difference in assumptions between Proposition 4 and Proposition 5 is the monotonicity of the function ϕ (κ). Furthermore, Step 1 in the proof of Proposition 4 does not use the monotonicity of the function ϕ (κ). Thus, for any policy g0 ∈ R, the pairs (θ, κ) ∈ R × R+ solving equations (43)-(44) in the paper are still given by: (1) the set of κ ∈ R+ satisfying the following condition ω (φc g0 + φλ )2 σ 2λ ln (2) ≤ ϕ (κ) with equality if κ > 0, and (2) the corresponding θ ∈ R given by the following equation ¢ ¡ 1 − 2−2κ (φ g0 + φλ ) . θ= 1 − (1 − φc ) (1 − 2−2κ ) c

(12)

(13)

Step 2: Optimal monetary policy has to satisfy g0 ≥ − φφλ . This step is identical to c

Step 2 in the proof of Proposition 4. The reason is again that Step 2 in the proof of Proposition 4 does not use the monotonicity of the function ϕ (κ). Thus, from now on, we can focus on policies g0 ≥ − φφλ . c

Step 3: Equilibrium price dispersion. This step is identical to Step 4 in the proof of Proposition 4. Step 4 in the proof of Proposition 4 derives equilibrium price dispersion without using the monotonicity of the function ϕ (κ). Hence, equilibrium price dispersion is given by i f 0 (κ) ¡1 − 2−2κ ¢ h 2 . (14) E (pi,t − pt ) = ω ln (2) Step 4: Equilibrium consumption. Substituting equation (13) into the equation for consumption ct = (g0 − θ) λt yields the following expression for equilibrium consumption " # ¢ ¡ 1 − 2−2κ (φc g0 + φλ ) ct = g0 − λt . 1 − (1 − φc ) (1 − 2−2κ )

(15)

Step 5: Optimal monetary policy when ϕ (κ) > ϕ (0) for all κ > 0. We now derive optimal monetary policy when the function ϕ (κ) satisfies ∀κ > 0 : ϕ (κ) > ϕ (0) .

6

(16)

To understand what this means in terms of primitives of the model, recall that £ ¤2 ϕ (κ) = f 0 (κ) φc 2κ + (1 − φc ) 2−κ .

First, we introduce notation. Let g¯0 denote the policy s ϕ (0) φλ 1 . g¯0 = − + φc φc ωσ 2λ ln (2)

(17)

i h For all policies g0 ∈ − φφλ , g¯0 , the unique equilibrium is κ = 0. For any policy g0 > g¯0 , κ = 0 is c

not an equilibrium. The first result follows from conditions (12) and (16), while the second result follows from condition (12) alone.

Second, consider the case of g¯0 ≥ 0. In this case, the unique optimal monetary policy is the inaction policy: g0∗ = 0. The reason is simple. The condition g¯0 ≥ 0 implies that at the inaction policy g0 = 0, price setters pay no attention to markup shocks. Formally, at the policy g0 = 0, the unique equilibrium is κ = 0, and thus price dispersion equals zero and consumption variance equals zero. The equilibrium allocation equals the efficient allocation. Furthermore, at any policy that differs from the inaction policy, the equilibrium allocation differs from the efficient allocation. If price setters pay attention to markup shocks, price dispersion is positive. If price setters pay no attention to markup shocks, consumption variance is positive since g0 6= 0. Hence, when g¯0 ≥ 0, the unique optimal monetary policy is the inaction policy: g0∗ = 0. Third, consider the case of g¯0 < 0. In this case, the unique optimal monetary policy is g0∗ = g¯0 . The easiest way to understand the proof is to consider a figure. Figure 1 depicts the function ϕ (κ). In Figure 1 we assume that the function ϕ (κ) has one local maximum and local minimum on R++ . Later we show that the proof extends to the case where the function ϕ (κ) has more than one local maximum and local minimum on R++ . The number of local maxima always equals the number of local minima because: (i) ϕ (0) < ϕ (κ) for all κ > 0, and (ii) ϕ (κ) diverges to infinity as κ goes to infinity. The horizontal lines represent the value of the left-hand side of condition (12) for different g0 ) represents the value of the left-hand side policies g0 ∈ R. For example, the horizontal line LHS (¯ of condition (12) at the policy g¯0 . For all policies g0 ≥ g¯0 , the set of equilibria for a given policy g0 is given by the intersection(s) of the horizontal line LHS (g0 ) and ϕ (κ). For example, at the policy g¯0 the unique equilibrium is point A; at the policy g00 the set of equilibria consists of points B and C; at the policy g000 the set of equilibria consists of points D, E, and F; and at the policy g0000 the set 7

h ´ of equilibria consists of points G and H. Finally, for all policies g0 ∈ − φφλ , g¯0 the equilibrium is a c ´ h φλ point on the vertical axis because for all policies g0 ∈ − φ , g¯0 the unique equilibrium is κ = 0. c

To prove that the unique optimal monetary policy is

g0∗

= g¯0 , we need to show that point A is

better than any other point that the central bank can achieve. First of all, we show that all policies h ´ g0 ∈ − φφλ , g¯0 are worse than the policy g0 = g¯0 , that is, all points on the vertical axis below point c i h A are worse than point A. For any policy g0 ∈ − φφλ , g¯0 the unique equilibrium is κ = 0, and thus c ´ h price dispersion equals zero. Consumption variance is strictly larger at a policy g0 ∈ − φφλ , g¯0 c ´ h than at the policy g0 = g¯0 because g¯0 < 0. Hence, any policy g0 ∈ − φφλ , g¯0 is worse than the c

policy g0 = g¯0 . Next, we show that all policies g0 > g¯0 are worse than the policy g0 = g¯0 if the cost

function for attention satisfies condition (46) in the paper on R++ . In other words, any point that the central bank can achieve above point A is also worse than point A. Recall that an increase in g0 shifts up the horizontal line LHS (g0 ). Let us focus first on price dispersion. Price dispersion at any policy g0 > g¯0 exceeds price dispersion at the policy g0 = g¯0 , because any policy g0 > g¯0 yields positive attention (κ > 0) whereas the policy g0 = g¯0 yields no attention (κ = 0). Let us turn to consumption variance. The policy g00 is the policy at which multiple equilibria start to exist, the policy g0000 is the policy at which multiple equilibria cease to exist, and the policy g000 is any policy g0 , g00 ), that is, any point between point between these two policies. Now consider any policy g0 ∈ (¯ A and point B on the graph of the function ϕ (κ). Since we have ϕ0 (κ) > 0 on this subset of R++ , the result of Step 6 in the proof of Proposition 4 applies. Thus, if the cost function for attention satisfies condition (46) in the paper on R++ , the fall in consumption after a positive markup shock in any point between points A and B is weakly larger than the fall in consumption after a positive markup shock in point A. Hence, any point between point A and point B is worse than point A. For the same reason, points B and D are worse than point A. Furthermore, point D can be moved arbitrarily close to point G. By continuity, point G is therefore also worse than point A. Hence, points B, D, and G are worse than point A. Moreover, point C is worse than point B, points E and F are worse than point D, and point H is worse than point G. The reason is that for any policy g0 ≥ − φφλ the increase in the price level after a positive markup shock is larger when attention is c

larger and thus the fall in consumption after a positive markup shock is larger when attention is larger. See equation (15). Hence, any point in the multiple equilibria region is worse than point A. Finally, any point above point H on the graph of the function ϕ (κ) is worse than point H.

8

The reason is again that the result of Step 6 in the proof of Proposition 4 applies since we have ϕ0 (κ) > 0 on this subset of R++ . In summary, if the cost function for attention satisfies condition (46) in the paper on R++ , any point that the central bank can achieve above point A is worse than point A, that is, any policy g0 > g¯0 is worse than the policy g0 = g¯0 . We already proved that in ´ h the case of g¯0 < 0 any policy g0 ∈ − φφλ , g¯0 is worse than the policy g0 = g¯0 . Combining results, c

we arrive at the following conclusion. If the cost function for attention satisfies condition (46) in the paper on R++ , then in the case of g¯0 < 0 the unique optimal monetary policy is g0∗ = g¯0 .

In Figure 1 we assume that the function ϕ (κ) has one local maximum and local minimum on R++ . The last sentence of the previous paragraph extends to the case where the function ϕ (κ) has more than one local maximum and local minimum on R++ . There are two possibilities: (i) adding a local maximum and local minimum creates a new multiple equilibria region, and (ii) adding a local maximum and local minimum does not create a new multiple equilibria region (but may increase the size of an existing multiple equilibria region). In either case, it is still true for any policy g0 > g¯0 that moving up on the graph of the function ϕ (κ) outside a multiple equilibria region reduces welfare and that the lower-left point of a multiple equilibria region is the best point in a multiple equilibria region. Step 6: Optimal monetary policy when ϕ (κ) ≤ ϕ (0) for some κ > 0. We now derive optimal monetary policy when the function ϕ (κ) satisfies ∃κ > 0 : ϕ (κ) ≤ ϕ (0) .

(18)

h ´ First, we introduce notation. Let gˆ0 denote the smallest g0 ∈ − φφλ , ∞ at which there exist c

multiple equilibria. It follows from condition (18) that gˆ0 ≤ g¯0 .

Second, consider the case of gˆ0 > 0. In this case, the unique optimal monetary policy is the inaction policy: g0∗ = 0. The reason is simple. The condition gˆ0 > 0 implies that at the inaction policy g0 = 0, price setters pay no attention to markup shocks. Formally, at the policy g0 = 0, the unique equilibrium is κ = 0, and thus price dispersion equals zero and consumption variance equals zero. The equilibrium allocation equals the efficient allocation. Furthermore, at any policy that differs from the inaction policy, the equilibrium allocation differs from the efficient allocation. If price setters pay attention to markup shocks, price dispersion is positive. If price setters pay no attention to markup shocks, consumption variance is positive since g0 6= 0. Hence, when gˆ0 > 0, the unique optimal monetary policy is the inaction policy: g0∗ = 0. 9

Third, consider the case of gˆ0 ≤ 0. In this case, the optimal monetary policy is a g0 marginally below gˆ0 . The easiest way to understand the proof is again to consider a figure. Figure 2 depicts the function ϕ (κ). In Figure 2 we assume that the function ϕ (κ) has the property (18) which is the case we are looking at now. In Figure 2 we also assume that the function ϕ (κ) has one local maximum and local minimum on R++ . Later we show that the proof extends to the case where the function ϕ (κ) has only a local minimum or more than one local maximum or local minimum on R++ . The horizontal lines represent the value of the left-hand side of condition (12) for different ´ h policies g0 ∈ R, as in Figure 1. For all policies g0 ∈ − φφλ , gˆ0 the equilibrium for a given policy is ´ hc a point on the vertical axis because for all policies g0 ∈ − φφλ , gˆ0 the unique equilibrium is κ = 0. c

For all policies g0 ≥ gˆ0 , the set of equilibria for a given policy consists of the intersection(s) of the

horizontal line LHS (g0 ) and ϕ (κ) as well as a point on the vertical axis if g0 ≤ g¯0 . The policy gˆ0 is the policy at which multiple equilibria start to exist, the policy g0000 denotes the policy at which multiple equilibria cease to exist, and the policy g000 denotes any policy between these two policies. At the policy gˆ0 the set of equilibria consists of points B and C; at the policy g000 the set of equilibria consists of points D, E, and F; and at the policy g0000 the set of equilibria consists of points G and H. By changing the policy g0 , the central bank can move the horizontal line LHS (g0 ) up or down and can thus achieve different outcomes. To derive optimal monetary policy, let us compare equilibria that the central bank can achieve to point B (i.e., the equilibrium with κ = 0 at the policy gˆ0 ). Consider first point C. Point C is worse than point B. Price dispersion is strictly larger in point C than in point B because κ > 0 in point C and κ = 0 in point B. The fall in consumption after a positive markup shock is also strictly larger in point C than in point B because nominal spending weakly falls after a positive markup shock in these two points (ˆ g0 ≤ 0) and the price level does not move in point B but strictly increases after a positive markup shock in point C (φc gˆ0 + φλ > 0 and κ > 0). Next, point F can be moved arbitrarily close to point C. By continuity, a point F arbitrarily close to point C is also worse than point B. Furthermore, if the cost function for attention satisfies condition (46) in the paper on R++ , any point to the right of point F on the graph of the function ϕ (κ) is worse than point F. Price dispersion is strictly larger in these points than in point F because attention is strictly larger. The fall in consumption after a positive markup shock is weakly larger in these points than in point F because ϕ0 (κ) > 0 on this subset of R++ and thus the result of Step 6 in the proof of Proposition 4 applies. Hence, points C, F, and H, and any point

10

to the right of point H on the graph of the function ϕ (κ) are worse than point B. Moreover, since points F and H are worse than point B also the worst point among points D, E, and F and the worst point among points G and H are worse than point B. We are now in the position to derive optimal monetary policy. We entertain two different assumptions about the central bank’s attitude towards multiple equilibria: (i) the central bank only considers policies g0 ∈ R that yield a unique equilibrium, and (ii) the central bank considers policies that yield multiple equilibria but evaluates them by looking at the worst equilibrium. Both assumptions yield the same result about optimal monetary policy. We begin with the assumption that the central bank only considers policies that / [ˆ g0 , g0000 ], that is, yield a unique equilibrium. Formally, the central bank only considers policies g0 ∈ the central bank only considers points below point B on the vertical axis and points above point H on the graph of the function ϕ (κ). If the cost function for attention satisfies condition (46) in the paper on R++ , points above point H on the graph of the function ϕ (κ) are worse than point B, and for these points, welfare is strictly decreasing in the distance to point H. This was shown above. Furthermore, points below point B on the vertical axis are also worse than point B, and for these points, welfare is strictly decreasing in the distance to point B. This follows from gˆ0 ≤ 0. Finally, the central bank can achieve welfare that is arbitrarily close to welfare in point B with a g0 marginally below gˆ0 , while the central bank cannot achieve welfare that is arbitrarily close to welfare in point B with a g0 marginally above g0000 . Hence, if the central bank only considers policies that yield a unique equilibrium and the cost function for attention satisfies condition (46) in the paper on R++ , the optimal monetary policy is a g0 marginally below gˆ0 . Complete price stabilization in response to markup shocks is optimal. Next, let us turn to the assumption that the central bank considers policies that yield multiple equilibria but evaluates them by looking at the worst equilibrium. If the central bank evaluates policies that yield multiple equilibria by looking at the worst equilibrium, any policy g0 ≥ gˆ0 yields welfare that is strictly smaller than welfare in point B. This was shown above. Furthermore, points below point B on the vertical axis are worse than point B, and for these points, welfare is strictly decreasing in the distance to point B. This was also pointed out above. Finally, the central bank can achieve welfare that is arbitrarily close to welfare in point B with a g0 marginally below gˆ0 , while the central bank cannot achieve welfare that is arbitrarily close to welfare in point B with a g0 ≥ gˆ0 if the central bank evaluates g0 , g0000 ] by looking at the worst equilibrium. Hence, if the central bank evaluates policies g0 ∈ [ˆ 11

policies that yield multiple equilibria by looking at the worst equilibrium and the cost function for attention satisfies condition (46) in the paper on R++ , the optimal monetary policy is again a g0 marginally below gˆ0 . Complete price stabilization in response to markup shocks is optimal. Finally, for completeness, let us entertain the assumption that the central bank evaluates policies that yield multiple equilibria by looking at the best equilibrium. In this case, if the cost function for attention satisfies condition (46) in the paper on R++ , the optimal monetary policy is ⎧ ⎨ 0 if g¯0 ≥ 0 . g0 = ⎩ g¯ if g¯ < 0 0 0 This policy is identical to the optimal monetary policy in Proposition 4 in the paper.

In Figure 2 we assume that the function ϕ (κ) has one local maximum and one local minimum on R++ . The results of the previous paragraph extend to the case where the function ϕ (κ) has no local maximum on R++ . In this case, point G simply coincides with point A. This does not change any of the arguments. Finally, the results of the previous paragraph also extend to the case where the function ϕ (κ) has more than one local maximum or more than one local minimum on R++ . Adding a local maximum and local minimum may or may not create a new multiple equilibria region. In either case, one simply has to use the fact that, if the cost function for attention satisfies condition (46) in the paper on R++ , moving up on the graph of the function ϕ (κ) outside a multiple equilibria region reduces welfare and the lower-left point of a new multiple equilibria region is the best point in a new multiple equilibria region.

12

C

Autocorrelated desired markup

In this section we discuss in detail results presented in Section 5.2 of the paper. When ρλ > 0 we solve for the optimal monetary policy numerically. This section describes the solution method, parameter values, and results.

C.1

Solution method

Let us begin with the Ramsey problem in the model with an exogenous information structure. We turn this infinite-dimensional problem into a finite-dimensional problem by restricting Gt (L) in equation (31) of the paper to be a time-invariant lag polynomial of an ARMA(2,2) process:1 mt = ρ1 mt−1 + ρ2 mt−2 + ϑ0 ν t + ϑ1 ν t−1 + ϑ2 ν t−2 . Following the procedure in Woodford (2002), one can then compute an exact linear rational expectations equilibrium of the model consisting of equations (22)-(33) in the paper for a given monetary policy by solving a Riccati equation. We then run a numerical optimization routine to obtain the optimal monetary policy. Let us turn to the Ramsey problem in the model with an endogenous information structure. We again turn this infinite-dimensional problem into a finite-dimensional problem by restricting Gt (L) to be a time-invariant lag polynomial of an ARMA(2,2) process. Following the procedure in Woodford (2002), one can then compute an exact linear rational expectations equilibrium of the model given by equations (22)-(33) of the paper for a given policy and a given signal precision. Furthermore, for a given law of motion for the endogenous variables, one can solve the attention problem given by equations (34)-(37) of the paper. Hence, solving for a linear rational expectations equilibrium of the rational inattention model for a given monetary policy amounts to solving a fixed point problem. We solve for the optimal monetary policy both by using an optimization routine and by evaluating the central bank’s objective for different policies on a fine grid, restricting the search to values of ρ1 and ρ2 that ensure the polynomial has both roots inside the unit circle and i h to values of ϑ0 , ϑ1 , and ϑ2 that are in the interval − φφλ , φφλ .2 c

1

c

We choose an ARMA(2,2) parameterization because it is well known from time series econometrics that an

ARMA(p,q) parameterization is a very flexible and parsimonious parameterization. 2 The analysis of optimal monetary policy in the case of an i.i.d. desired markup suggests that the optimal values

13

C.2

Parameter values

Our benchmark parameter values are: β = 0.99, γ = 1, ψ = 0, α = 1, Λ = 1/4, ρλ = 0.9, and σ ν = 0.05. The value for β is a standard value for a quarterly model. The values for γ, ψ, and α are standard values in the business cycle literature. Similar values have been used, for instance, by Golosov and Lucas (2007) in the analysis of impulse responses to aggregate shocks. The value for Λ implies a steady state price elasticity of demand of five. This value for the price elasticity of demand is half way between the value of three used by Midrigan (2011) and the value of seven used by Golosov and Lucas (2007).3 The values ρλ = 0.9 and σ ν = 0.05 are obtained from estimates of the markup process by Smets and Wouters (2007).4 This set of parameter values has the property that there is no strategic complementarity in price setting: φc = 1. This implies that the equilibrium is unique for any given monetary policy. We also solved the two Ramsey problems for other parameter values of interest: α = 2/3 (so that the steady state labor share is 2/3), ψ = 1 (so that the Frisch elasticity of labor supply is 1/2), and Λ = 1/7 (so that the steady state markup is 14 percent). We never found multiple equilibria at a given monetary policy and we always obtained the same qualitative results regarding optimal monetary policy. In this online Appendix C there are no technology shocks because in Section 5 of the paper there are no technology shocks. However, in some later appendices there are technology shocks. For easy reference, we already state here the technology parameters that we will assume in those appendices. We choose the parameters of the AR(1) process for aggregate technology to match U.S. quarterly TFP growth estimated by Fernald (2009). Namely, we set ρa = 0.95 and σ a = 0.0085. of ϑ0 , ϑ1 , and ϑ2 should be in this interval. We have extended our search also to values outside this interval but we l k always found the optimal ϑ0 , ϑ1 , and ϑ2 to be within − φφλ , φφλ . c c 3 A price elasticity of demand of five is towards the upper end of estimates in the empirical IO literature. 4 In Smets and Wouters (2007), the markup process (“εpt ”) enters equation (10): πt = π1 πt−1 + π2 Et π t+1 −   Λ εpt . They assume that the markup process is an ARMA(1,2) process: εpt = ρp εpt−1 + σp η pt − μp η pt−1 , π3 μpt + π 3 1+Λ where η pt is i.i.d. standard normal. The mode estimates of Smets and Wouters (2007) are such that ρp = 0.9,

μp = 0.74, and σp =

1+Λ π3 Λ

∗ 0.14%. We assume an AR(1) process for the desired markup and our parameter choices

imply a different value of π 3 (namely, π 3 = 0.19). We set the autocorrelation of the desired markup to ρλ = 0.9 (i.e., equal to ρp in Smets and Wouters (2007)). We set the standard deviation of the innovation in the desired markup so that the standard deviation of the desired markup in our paper equals the standard deviation of εpt in Smets and Wouters (2007), after adjusting for the different value of π3 . This yields σν = 0.05.

14

C.3

Results for the benchmark signal structure

The lower panel of Figure 3 shows the optimal monetary policy response to a markup shock in the model with an exogenous information structure when ρλ = 0.9. For comparison, the upper panel of Figure 3 shows optimal monetary policy when ρλ = 0. Figure 3 also shows the optimal monetary policy in the Calvo model. All impulse responses are to a positive one standard deviation markup shock. A response equal to one means a one percent deviation from the non-stochastic steady state. Time is measured in quarters along the horizontal axis. In the Calvo model decision-makers in firms have perfect information and in every period each firm can adjust its price with an exogenous probability. Following Nakamura and Steinsson (2008), we set the fraction of firms that can adjust their price in a quarter equal to 0.4. The standard deviation of noise in the model with exogenous noisy signals is set such that the model with exogenous noisy signals and the Calvo model yield the same impulse response of the price level to a markup shock when the component of the profit-maximizing price driven by markup shocks follows a random walk. The idea is that we want to compare these two models for parameter values that imply the same degree of stickiness of the price level. The first result is the following. In the model with exogenous noisy signals, complete price stabilization in response to markup shocks is suboptimal when ρλ > 0. At the optimal monetary policy, the price level strictly increases on impact of a positive markup shock. Figure 3 shows this result for our benchmark parameter values. We solved the Ramsey problem for many different sets of parameter values with ρλ > 0 and we always obtained this result. Comparing the optimal monetary policy response to markup shocks in the model with exogenous noisy signals and in the Calvo model, we obtain two results. First, the optimal monetary policy is qualitatively similar in these two models. At the optimal monetary policy, the price level strictly increases on impact of a positive markup shock, the consumption level strictly falls on impact of a positive markup shock, and there is inefficient price dispersion. Second, whether optimal monetary policy is also quantitatively similar in these two models depends on parameter values. When ρλ = 0.9, the impulse responses at the optimal monetary policy are very similar in the two models. In contrast, when ρλ = 0, the optimal policy in the model with exogenous noisy signals is to respond to the markup shock only in the period of the shock, while the optimal policy in the Calvo model is to respond to the markup shock also in the periods after the shock. 15

To summarize, in the model with exogenous noisy signals, there exists a trade-off between reducing inefficient price dispersion and reducing inefficient consumption variance and complete price stabilization in response to markup shocks is suboptimal. Furthermore, optimal monetary policy is qualitatively similar to optimal monetary policy in the Calvo model. Next, consider the model with an endogenous information structure. We consider two cost functions for attention, f (κ) = μκ and f (κ) = μ22κ , and we solve the Ramsey problem for values ¤ £ of (μ/ω) ∈ 10−5 , 1 . We find that complete price stabilization in response to markup shocks is

optimal. We solved the Ramsey problem for many different sets of parameter values with ρλ > 0 and we always obtained this result.5 Figure 4 depicts the optimal monetary policy for f (κ) = μκ and (μ/ω) = 10−4 for our bench-

mark parameter values. The figure shows the impulse responses of nominal spending, the consumption level, the price level, and the profit-maximizing price to a one standard deviation positive markup shock. The optimal monetary policy is to reduce nominal spending on impact of a positive markup shock so as to counteract the effect of the markup shock on the profit-maximizing price. At the optimal monetary policy, price setters pay no attention to random variation in the desired markup (i.e., κ∗ = 0) and therefore the price level does not respond to markup shocks. Complete price stabilization in response to markup shocks is optimal.

C.4

Results for the alternative signal structure

So far we have assumed that price setters receive signals of the form “desired markup plus i.i.d. noise.” In other words, price setters receive signals about the fundamental with noise that is independent over time. In Section 5.1, a signal of the form “desired markup plus i.i.d. noise” is optimal.6 The reason is that the desired markup follows a white noise process and the profit-maximizing price depends only on the current value of the desired markup. However, in Section 5.2 where ρλ > 0, a signal of the form “desired markup plus i.i.d. noise” is not necessarily optimal. One could in principle solve for the optimal form of the signal within the attention problem for a given monetary policy. Unfortunately, the time required to solve for equilibrium at a given monetary policy increases substantially and thus the optimal monetary policy problem becomes 5

We solved the Ramsey problem for many different values of (μ/ω) between 10−5 and 1. We also solved the

Ramsey problem for ψ = 1 and α = 2/3. 6 This follows from Propositions 1-4 in Mackowiak and Wiederholt (2009).

16

computationally much harder to solve. Therefore, we decided to assess robustness of the results presented in Section 5.2 by studying optimal monetary policy for an alternative signal structure. In the benchmark model setup, price setters receive signals of the form “desired markup plus i.i.d. noise.” Now we instead assume that price setters receive signals of the form “profit-maximizing price plus i.i.d. noise.” Otherwise the Ramsey problem (21)-(37) remains unchanged. We chose this signal structure because we know this signal structure is optimal when the profit-maximizing price follows an AR(1) process.7 For the alternative signal structure, we solved the Ramsey problem for two cost functions for £ ¤ attention, f (κ) = μκ and f (κ) = μ22κ , and many different parameter values (μ/ω) ∈ 10−5 , 1 . We always obtained the result that complete price stabilization in response to markup shocks is

optimal. At the optimal monetary policy, price setters pay no attention to markup shocks (i.e., κ∗ = 0) and therefore the price level does not respond to markup shocks. Figure 5 shows an example. The figure depicts the optimal monetary policy for f (κ) = μκ, (μ/ω) = 10−4 , and our benchmark parameter values. The figure shows the impulse responses of nominal spending, the consumption level, the price level, and the profit-maximizing price to a one standard deviation positive markup shock. Note that the central bank reduces nominal spending on impact of a positive markup shock and thereby stabilizes the profit-maximizing price in response to markup shocks. At the optimal monetary policy, price setters pay no attention to random variation in the desired markup. 7

This follows from Propositions 1-4 in Mackowiak and Wiederholt (2009). However, we do not know whether this

signal structure is optimal when the profit-maximizing price does not follow an AR(1) process.

17

D

Correlation in noise across firms

In this section we discuss in detail results presented in Section 6.3 of the paper.

D.1

Assumptions

We assume that the noise in the signal in equation (28) of the paper has the following properties: © ª∞ © ª∞ ∞ (i) the processes η i,t t=0 and ζ i,t t=0 are independent of the processes {At }∞ t=0 and {Λt }t=0 , (ii) © ª∞ © ª∞ the processes η i,t t=0 and ζ i,t t=0 are independent of each other and correlated across firms with

correlation coefficients ρη ∈ [0, 1] and ρζ ∈ [0, 1], and (iii) η i,t and ζ i,t follow Gaussian white noise

processes with variances σ 2η and σ 2ζ . Otherwise we do not change the Ramsey problem given by equations (21)-(37) in the paper. The monetary policy is given by equation (31) in the paper, and thus nominal spending does not respond to the firms’ information-processing noise. When the signal concerning aggregate technology has positive precision, one can write the noise in this signal without loss in generality as η i,t = σ η

hp i q ρη η¯t + 1 − ρη η˜i,t ,

where η¯t and η˜i,t follow independent Gaussian white noise processes with unit variance, and the term η¯t is common to all firms, while the term η˜i,t is independent across firms. When the signal concerning aggregate technology has zero precision, the signal contains no information about i h p √ ρη η¯t + 1 − ρη η˜i,t . aggregate technology and price setters observe perfectly the noise term Similarly, when the signal concerning the desired markup has positive precision, one can write

the noise in this signal without loss in generality as ζ i,t = σ ζ

hp i q ρζ ζ¯t + 1 − ρζ ζ˜i,t ,

where ζ¯t and ζ˜i,t follow independent Gaussian white noise processes with unit variance, and the term ζ¯t is common to all firms, while the term ζ˜i,t is independent across firms. When the signal concerning the desired markup has zero precision, the signal contains no information about the i h p √ ¯ ρζ ζ t + 1 − ρζ ζ˜i,t . desired markup and price setters observe perfectly the noise term

18

D.2

Optimal monetary policy response to technology shocks

Proposition 2 Consider the Ramsey problem (21)-(37) in the paper. In contrast to the paper, assume that the noise η i,t is correlated across firms with any correlation coefficient ρη ∈ (0, 1]. Assume that σ 2λ = λ−1 = 0, σ 2a > 0, and ρa = 0. Consider policies of the form mt = f0 at and equilibria of the form pt = θa at + θη¯ η¯t . The unique optimal monetary policy response to aggregate technology shocks is mt =

φa at . φc

At the optimal monetary policy, the price level does not respond to aggregate technology shocks and the equilibrium allocation equals the efficient allocation. Proof. By setting mt =

φa φc at

the central bank can completely offset the effect of technology

shocks on the profit-maximizing price. Equation (39) in the paper reduces to p∗i,t = (1 − φc ) pt and the unique equilibrium is: pt = 0, ct =

φa φc at ,

and pi,t − pt = 0. The equilibrium allocation equals

the efficient allocation. Furthermore, any other policy does not achieve the efficient allocation. If at the policy mt 6=

φa φc at

price setters pay attention to aggregate technology shocks, the correlated

noise in the price setters’ signals causes movements in the price level, which causes inefficient consumption fluctuations. If at the policy mt 6=

φa φc at

price setters pay no attention to aggregate

technology shocks, the consumption response to aggregate technology shocks is inefficient because ct = mt 6=

D.3

φa φc at .

Hence, the unique optimal monetary policy is mt =

φa φc at .

Optimal monetary policy response to markup shocks

Proposition 3 Consider the Ramsey problem (21)-(37) in the paper. In contrast to the paper, assume that the noise ζ i,t is correlated across firms with any correlation coefficient ρζ ∈ (0, 1]. Assume σ 2a = a−1 = 0, σ 2λ > 0, and ρλ = 0. Furthermore, to shorten the proof, assume φc ∈ (0, 1], that is, prices are not strategic substitutes. Consider policies of the form mt = g0 λt and equilibria of £ ¡ ¢ ¤2 ˜ (κ) = f 0 (κ) φc 2κ + (1 − φc ) 1 − ρζ 2−κ the form pt = θλ λt + θ¯ζ ζ¯t . Suppose that the function ϕ is strictly increasing on R+ and has no saddle point on R++ . Finally, define two policies, s f 0 (0) φλ φλ , g¯0 = − + φc φc ωφ2λ σ 2λ ln (2)

19

and φ φ gˆ0 = − λ + λ φc φc

s

¤ £ f 0 (0) 1 − (1 − φc ) ρζ ωφ2λ σ 2λ ln (2)

.

Note that gˆ0 ≤ g¯0 , with equality if and only if (1 − φc ) ρζ = 0. Then, the following results hold. First, optimal monetary policy has to satisfy φλ . φc i h Second, there are three regions: (i) if g0 ∈ − φφλ , gˆ0 there exists a unique equilibrium and κ = 0, g0 ≥ − c

g0 , g¯0 ] there exist two equilibria, one with κ = 0 and another with κ > 0, and (iii) if (ii) if g0 ∈ (ˆ

g0 > g¯0 there exists a unique equilibrium and κ > 0. Third, assume that the central bank evaluates policies that yield multiple equilibria by looking at the worst equilibrium or that the central bank only considers policies that yield a unique equilibrium. If the function f (κ) satisfies condition (46) in the paper on R++ , the unique optimal monetary policy is ⎧ ⎨ gˆ0 if gˆ0 < 0 g0∗ = . ⎩ 0 otherwise

At this policy, price setters pay no attention to markup shocks, the price level does not respond to markup shocks, price dispersion equals zero, and consumption variance equals g02 σ 2λ . Proof. Step 1: Equilibrium for a given policy. The profit-maximizing price and the actual price of good i in period t equal p∗i,t = [(1 − φc ) θλ + φc g0 + φλ ] λt + (1 − φc ) θζ¯ ζ¯t ,

where

(19)

£ ¤ pi,t = [(1 − φc ) θλ + φc g0 + φλ ] E [λt |Ii,t ] + (1 − φc ) θζ¯ E ζ¯t |Ii,t , E [λt |Ii,t ] =

and £ ¤ E ζ¯t |Ii,t =

⎧ ⎪ ⎨

⎧ ⎨ ⎩

σ 2λ σ 2λ +σ2ζ

0

¡ ¢ λt + ζ i,t if

σ 2λ σ 2ζ

>0

,

(21)

otherwise

¢ σ2 λt + ζ i,t if σλ2 > 0 ζ . ³ ´ p ⎪ ⎩ √ρζ √ρζ ζ¯t + 1 − ρζ ζ˜i,t otherwise √ σζ ρζ 2 σλ +σ2ζ

(20)

¡

(22)

Furthermore, equation (36) in the paper implies

σ 2λ = 22κ − 1. σ 2ζ 20

(23)

Next, we derive the equilibrium price level for a given allocation of attention. When σ 2λ /σ 2ζ > 0, it follows from equations (20)-(22) and equation (24) in the paper that the price level in period t equals pt

"

√ # σ ζ ρζ σ 2λ = [(1 − φc ) θλ + φc g0 + φλ ] 2 + (1 − φc ) θ¯ζ 2 λt σ λ + σ 2ζ σ λ + σ 2ζ " √ # σ ζ ρζ σ 2λ p σ ζ ρζ ζ¯t . + [(1 − φc ) θλ + φc g0 + φλ ] 2 + (1 − φc ) θ¯ζ 2 2 2 σλ + σζ σλ + σζ

Thus, for a given allocation of attention, the unique rational expectations equilibrium of the form pt = θλ λt + θ¯ζ ζ¯t is given by σ2

λ (φc g0 + φλ ) σ2 +σ 2 λ ζ θλ = , ¢ σ2λ ¡ 1 − (1 − φc ) 1 − ρζ σ2 +σ2 − (1 − φc ) ρζ λ

and

θ¯ζ = θλ σ ζ

ζ

p ρζ .

Furthermore, when σ 2λ /σ 2ζ = 0, it follows from equations (20)-(22) and equation (24) in the paper that the price level in period t equals pt = (1 − φc ) θζ¯ ρζ ζ¯t . Since (1 − φc ) ρζ 6= 1, the unique rational expectations equilibrium of the form pt = θλ λt + θ¯ζ ζ¯t is θλ = θζ¯ = 0. Hence, for any given allocation of attention κ ≥ 0, the unique rational expectations equilibrium of the form pt = θλ λt + θ¯ζ ζ¯t is given by

and

¡ ¢ (φc g0 + φλ ) 1 − 2−2κ ¡ ¢ θλ = , 1 − (1 − φc ) 1 − ρζ (1 − 2−2κ ) − (1 − φc ) ρζ ⎧ q ρζ σ2λ ⎨ θ λ 22κ −1 θ¯ζ = ⎩ 0

where we have used equation (23).

21

if κ > 0 otherwise

,

(24)

(25)

We now turn to the optimal allocation of attention of an individual firm. Equations (19)-(20) imply E

i h h¡ ¢2 i = [(1 − φc ) θλ + φc g0 + φλ ]2 E (λt − E [λt |Ii,t ])2 pi,t − p∗i,t h¡ ¤¢2 i £ + (1 − φc )2 θ2¯ζ E ζ¯t − E ζ¯t |Ii,t £ ¡ £ ¤¢¤ +2 [(1 − φc ) θλ + φc g0 + φλ ] (1 − φc ) θζ¯ E (λt − E [λt |Ii,t ]) ζ¯t − E ζ¯t |Ii,t .

Furthermore, one can show that

i h E (λt − E [λt |Ii,t ])2 = 2−2κ σ 2λ , E and

h¡ £ ¤¢2 i ζ¯t − E ζ¯t |Ii,t = 1 − 2−2κ ρζ ,

p ¤¢¤ ¡ £ £ p = − 2−2κ − 2−4κ σ λ ρζ . E (λt − E [λt |Ii,t ]) ζ¯t − E ζ¯t |Ii,t

Substituting the last three equations into the previous equation yields E

h¡ ¢2 i = [(1 − φc ) θλ + φc g0 + φλ ]2 2−2κ σ 2λ pi,t − p∗i,t ¡ ¢ + (1 − φc )2 θ2¯ζ 1 − 2−2κ ρζ

−2 [(1 − φc ) θλ + φc g0 + φλ ] (1 − φc ) θ¯ζ

p p 2−2κ − 2−4κ σ λ ρζ .

(26)

When θλ = θ¯ζ = 0 or φc = 1, the optimal allocation of attention is given by o 1 nω (φc g0 + φλ )2 σ 2λ 2−2κ + f (κ) . κ∈R+ 1 − β 2 min

In this case the correlated noise is not of interest to the firm because the correlated noise does not appear in the price level or the price level does not affect the profit-maximizing price. The solution to the last attention problem is ⎧ 2 2 0 ⎨ κ f oc if ω (φc g0 + φλ ) σ λ ln (2) > f (0) , κ= ⎩ 0 otherwise

(27)

where κf oc is defined implicitly by

ω (φc g0 + φλ )2 σ 2λ ln (2) = f 0 (κf oc ) 22κf oc .

22

(28)

When (i) θλ 6= 0, (ii) θλ and θ¯ζ are given by equations (24)-(25), and (iii) φc 6= 1, the optimal allocation of attention is given by a slightly more complicated argument. One can show that (i)-(iii) √ imply that θζ¯ 6= 0 and [(1 − φc ) θλ + φc g0 + φλ ] (1 − φc ) θ¯ζ σ λ ρζ > 0. Furthermore, when θλ 6= 0, √ θζ¯ 6= 0, φc 6= 1, and [(1 − φc ) θλ + φc g0 + φλ ] (1 − φc ) θ¯ζ σ λ ρζ > 0, the marginal benefit of paying attention goes to infinity as κ goes to zero. Hence, it is optimal to pay at least some attention and the optimal allocation of attention is given by the first-order condition ω [(1 − φc ) θλ + φc g0 + φλ ]2 σ 2λ ln (2) −ω (1 − φc )2 θ2¯ζ ρζ ln (2) −ω [(1 − φc ) θλ + φc g0 + φλ ] (1 − φc ) θ¯ζ σ λ = f 0 (κ) 22κ .

p 22κ − 2 ρζ ln (2) √ 22κ − 1

(29)

The optimal allocation of attention of an individual firm is unique (given the behavior of other firms, that is, given θλ and θ¯ζ ). The reason is simple. Both sides of the last equation are continuous functions of κ on R++ ; the left-hand side is a strictly decreasing function of κ on R++ which goes to plus infinity as κ goes to zero and eventually becomes negative as κ increases; and the right-hand side is a strictly increasing function of κ on R++ which goes to f 0 (0) ≥ 0 as κ goes to zero. Finally, equations (27)-(29) cover all the relevant cases because in an equilibrium either θλ = θ¯ζ = 0 or θλ 6= 0 and θλ and θ¯ζ are given by equations (24)-(25).

We are now in the position to characterize the set of equilibria of the form pt = θλ λt + θζ¯ ζ¯t at

a given policy g0 ∈ R. First, we derive the condition for existence of an equilibrium with κ = 0. Equations (24)-(25) imply that when κ = 0 then θλ = θ¯ζ = 0. Furthermore, equations (27)-(28) imply that when θλ = θζ¯ = 0 then κ = 0 is the optimal allocation of attention if and only if ω (φc g0 + φλ )2 σ 2λ ln (2) ≤ f 0 (0) .

(30)

Hence, there exists a rational expectations equilibrium of the form pt = θλ λt + θ¯ζ ζ¯t with κ = 0 if and only if condition (30) is satisfied. Second, we derive the condition for existence of an equilibrium with κ > 0. Note that in an equilibrium with κ > 0, one cannot have θλ = θ¯ζ = 0. The reason is simple. If κ > 0, then θλ = θζ¯ = 0 implies (φc g0 + φλ ) = 0. See equations (24)-(25). Furthermore, when θλ = θ¯ζ = (φc g0 + φλ ) = 0, the optimal allocation of attention is κ = 0. See equation (27). Thus, in an 23

equilibrium with κ > 0, one must have: θλ 6= 0 and θλ and θ¯ζ are given by equations (24)-(25). The optimal allocation of attention is then given by equation (28) if φc = 1 and by equation (29) if φc 6= 1. Since these two equations are identical when φc = 1, one can just focus on equation (29). Substituting equations (24)-(25) for κ > 0 into equation (29) and rearranging yields ω (φc g0 + φλ )2 σ 2λ ln (2) =

£ ¡ ¢ ¤2 1 f 0 (κ) φc 2κ + (1 − φc ) 1 − ρζ 2−κ . 1 − (1 − φc ) ρζ

(31)

Note that the right-hand side of the last equation is a continuous function of κ on R+ . We £ ¡ ¢ ¤2 arrive at the following conclusion. If ϕ ˜ (κ) ≡ f 0 (κ) φc 2κ + (1 − φc ) 1 − ρζ 2−κ is a strictly increasing function of κ on R+ , then: (i) there exists a rational expectations equilibrium of the form pt = θλ λt + θ¯ζ ζ¯t with κ > 0 if and only if £ ¤ ω (φc g0 + φλ )2 σ 2λ ln (2) > f 0 (0) 1 − (1 − φc ) ρζ ,

(32)

and (ii) there exists at most one equilibrium with κ > 0 and this equilibrium is given by the solution to equation (31). In summary, there are three regions: £ ¤ • If ω (φc g0 + φλ )2 σ 2λ ln (2) ≤ f 0 (0) 1 − (1 − φc ) ρζ , there exists a unique equilibrium: κ = 0.

¡ £ ¤ ¤ • If ω (φc g0 + φλ )2 σ 2λ ln (2) ∈ f 0 (0) 1 − (1 − φc ) ρζ , f 0 (0) , there exist two equilibria: κ = 0 and the κ > 0 solving equation (31).

• If ω (φc g0 + φλ )2 σ 2λ ln (2) > f 0 (0), there exists a unique equilibrium: the κ > 0 solving equation (31). Step 2: Optimal monetary policy has to satisfy g0 ≥ − φφλ . The equations ct = mt − pt , c

mt = g0 λt , and pt = θλ λt + θ¯ζ ζ¯t , and the independence of λt and ζ¯t imply that £ ¤ E c2t = (g0 − θλ )2 σ 2λ + θ2¯ζ .

(33)

In the following, we call the first term on the right-hand side “fundamental consumption variance” and the second term on the right-hand side “non-fundamental consumption variance.” At the i h monetary policy g0 = − φφλ , the unique equilibrium is κ = 0, implying E (pi,t − pt )2 = 0 and c ³ ´2 £ 2¤ φλ E ct = φ σ 2λ . Next, consider a monetary policy g0 < − φφλ . Price dispersion and nonc

c

fundamental consumption variance are weakly larger at the policy g0 < − φφλ than at the policy c

24

g0 = − φφλ , because they are both non-negative in general and zero at the policy g0 = − φφλ . c

c

Furthermore, fundamental consumption variance is strictly larger at the policy g0 < the policy g0 =

− φφλ , c

because for all g0 <

− φφλ c

− φφλ c

than at

we have ¡ ¢ (φc g0 + φλ ) 1 − 2−2κ φ ¡ ¢ < − λ < 0. g0 − θλ = g0 − −2κ φc φc + (1 − φc ) 1 − ρζ 2

Hence, a monetary policy g0 < − φφλ cannot be optimal. From now on, we focus on policies c

g0 ≥ − φφλ . c

Step 3: Equilibrium attention as a function of policy. If ω (φc g0 + φλ )2 σ 2λ ln (2) > £ ¤ f 0 (0) 1 − (1 − φc ) ρζ , there exists an equilibrium with κ > 0 and equilibrium attention in this

equilibrium is given by equation (31). We now study how equilibrium attention in an equilibrium with κ > 0 varies with the policy rule (i.e., with the value of g0 ). The implicit function theorem

and ϕ ˜ 0 (κ) > 0 yield

ω2 (φc g0 + φλ ) φc σ 2λ ln (2) ∂κ = > 0. 1 ∂g0 ˜ 0 (κ) 1−(1−φ )ρ ϕ c

(34)

ζ

Step 4: Equilibrium price dispersion as a function of policy. If price setters pay no attention to markup shocks (κ = 0), price dispersion equals zero. If price setters do pay attention to markup shocks (κ > 0), price dispersion may be positive. More precisely, when price setters pay attention to markup shocks, the price of good i equals

the relative price of good i equals

¡ ¢ pi,t = θλ λt + ζ i,t , pi,t − pt = θλ σ ζ

and price dispersion equals

q 1 − ρζ ζ˜i,t ,

i h ¡ ¢ E (pi,t − pt )2 = θ2λ σ 2ζ 1 − ρζ .

Substituting equations (23) and (24) into the last equation yields ¡ ¢2 i h ¢ (φc g0 + φλ )2 1 − 2−2κ σ2 ¡ 2 1 − ρζ . E (pi,t − pt ) = £ ¡ ¢ ¤2 2κ λ φc + (1 − φc ) 1 − ρζ 2−2κ 2 − 1 Finally, substituting equation (31) into the last equation and rearranging yields ¡ ¢ h i 1 − ρζ f 0 (κ) 1 − 2−2κ 2 . E (pi,t − pt ) = 1 − (1 − φc ) ρζ ω ln (2) 25

(35)

Hence, in an equilibrium with κ > 0, equilibrium price dispersion is a strictly increasing function of equilibrium attention if ρζ ∈ (0, 1), while equilibrium price dispersion equals zero if ρζ = 1. Step 5: Equilibrium consumption variance as a function of policy. Consumption variance is given by £ ¤ E c2t = (g0 − θλ )2 σ 2λ + θ2¯ζ .

See equation (33). We refer to the first term on the right-hand side of the last equation as “fundamental consumption variance” and the second term as “non-fundamental consumption variance.” Consider first non-fundamental consumption variance. In an equilibrium with κ = 0, we have θζ¯ = 0 and thus non-fundamental consumption variance equals zero. See equation (25). In an equilibrium with κ > 0, equations (24)-(25) imply ¡ ¢2 ρζ σ 2λ (φc g0 + φλ )2 1 − 2−2κ 2 . θζ¯ = £ ¢ ¤2 2κ ¡ φc + (1 − φc ) 1 − ρζ 2−2κ 2 − 1 Substituting equation (31) into the last equation yields θ2ζ¯

¢ ¡ ρζ f 0 (κ) 1 − 2−2κ . = 1 − (1 − φc ) ρζ ω ln (2)

(36)

Hence, in an equilibrium with κ > 0, non-fundamental consumption variance is a strictly increasing function of equilibrium attention. (Comparing equations (35) and (36), note that equilibrium price dispersion is proportional to non-fundamental consumption variance.) Step 6: Equilibrium price level as a function of policy. The price level equals pt = θλ λt + θ¯ζ ζ¯t , where θλ is given by equation (24). In an equilibrium with κ = 0, we have θλ = 0. Furthermore, in an equilibrium with κ > 0, we have θλ > 0 and equation (24) implies £ ¤ ¡ ¢ 1 − (1 − φc ) ρζ 2−2κ 2 ln (2) ∂κ 1 − 2−2κ ∂θλ ¡ ¢ =£ (φc g0 + φλ ) + φ, ¡ ¢ ¤2 ∂g0 φc + (1 − φc ) 1 − ρζ 2−2κ c φc + (1 − φc ) 1 − ρζ 2−2κ ∂g0 where

∂κ ∂g0

is given by equation (34). The first term is the attention effect that is only present in

the model with an endogenous information structure. The second term is the usual effect which is also present in the model with an exogenous information structure. Substituting equation (34) into the last equation yields ¤ £ 1 − (1 − φc ) ρζ 2−2κ 2 ln (2) ω2φc σ 2λ ln (2) ∂θλ (φc g0 + φλ )2 = £ ¡ ¢ ¤2 1 ∂g0 ˜ 0 (κ) φc + (1 − φc ) 1 − ρζ 2−2κ 1−(1−φc )ρζ ϕ ¡ ¢ 1 − 2−2κ ¡ ¢ + φ. φc + (1 − φc ) 1 − ρζ 2−2κ c 26

Furthermore, substituting equation (31) into the last equation yields £ ¤ ¡ ¢ 1 − (1 − φc ) ρζ 4 ln (2) φc f 0 (κ) 1 − 2−2κ ∂θλ ¡ ¢ + = φ. ∂g0 ϕ ˜ 0 (κ) φc + (1 − φc ) 1 − ρζ 2−2κ c

Rearranging the right-hand side of the last equation yields £ ¤ 1 − (1 − φc ) ρζ 4 ln (2) φc f 0 (κ) 1 − (1 − φc ) ρζ ∂θλ ¡ ¢ +1 = − 0 2κ ∂g0 ϕ ˜ (κ) φc 2 + (1 − φc ) 1 − ρζ ¢¤ £ ¡ ˜ 0 (κ) 4 ln (2) φc f 0 (κ) φc 22κ + (1 − φc ) 1 − ρζ − ϕ £ ¡ ¢¤ = + 1. 1 ˜ 0 (κ) φc 22κ + (1 − φc ) 1 − ρζ 1−(1−φ )ρ ϕ c

ζ

The definition of the function ϕ ˜ (κ) implies

£ ¡ ¢ ¤2 ϕ ˜ 0 (κ) = f 00 (κ) φc 2κ + (1 − φc ) 1 − ρζ 2−κ ¢ ¤£ ¢ ¤ £ ¡ ¡ +f 0 (κ) 2 φc 2κ + (1 − φc ) 1 − ρζ 2−κ φc 2κ − (1 − φc ) 1 − ρζ 2−κ ln (2) .

Substituting the last equation into the numerator of the previous equation and rearranging yields ¡ ¢ −2κ ¤ + (1 − φ ) 1 − ρ 2 φ ∂θλ £ 0 c c ζ = f (κ) 2 ln (2) − f 00 (κ) + 1. 1 0 ∂g0 ˜ (κ) 1−(1−φ )ρ ϕ c

ζ

Hence, in an equilibrium with κ > 0 and when ϕ ˜ 0 (κ) > 0, we have

∂θλ ∂g0

≥ 1 if and only if

f 00 (κ) ≤ 2 ln (2) . f 0 (κ)

(37)

Step 7: Optimal monetary policy. To derive optimal policy, we first define two policies. Let g¯0 denote the policy g0 ≥ − φφλ satisfying c

ω (φc g0 + φλ )2 σ 2λ ln (2) = f 0 (0) . Let gˆ0 denote the policy g0 ≥ − φφλ satisfying c

£ ¤ ω (φc g0 + φλ )2 σ 2λ ln (2) = f 0 (0) 1 − (1 − φc ) ρζ .

h i The policies g¯0 and gˆ0 are important for the following reason. For all g0 ∈ − φφλ , g¯0 , there exists an c

equilibrium with κ = 0. For all g0 > gˆ0 , there exists an equilibrium with κ > 0. See step 1. When gˆ0 = g¯0 (that is, (1 − φc ) ρζ = 0), there always exists a unique equilibrium. When gˆ0 < g¯0 (that is, g0 , g¯0 ] and a unique equilibrium otherwise. (1 − φc ) ρζ > 0) there exist two equilibria for all g0 ∈ (ˆ In an equilibrium with κ = 0, price dispersion equals zero, non-fundamental consumption variance 27

equals zero, and fundamental consumption variance equals (g0 − θλ )2 σ 2λ = g02 σ 2λ . In an equilibrium with κ > 0, equilibrium attention is given by equation (31), the price level response to markup shocks θλ is given by equation (24), price dispersion is given by equation (35), non-fundamental consumption variance is given by equation (36), and fundamental consumption variance equals (g0 − θλ )2 σ 2λ . We now go separately over three cases: (1) gˆ0 ≥ 0, (2) gˆ0 < 0, g¯0 ≤ 0, and (3) gˆ0 < 0, g¯0 > 0.

£ ¤ Consider first the case gˆ0 ≥ 0, that is, ωφ2λ σ 2λ ln (2) ≤ f 0 (0) 1 − (1 − φc ) ρζ . In this case,

at the policy g0 = 0, the unique equilibrium allocation of attention is κ = 0, implying that price dispersion equals zero, non-fundamental consumption variance equals zero, and fundamental

consumption variance equals zero since g02 σ 2λ = 0. The policy g0 = 0 achieves the efficient allocation. Furthermore, any other policy does not achieve the efficient allocation. If at the policy g0 6= 0 price setters still pay no attention to markup shocks, fundamental consumption variance is positive because g02 σ 2λ > 0. If at the policy g0 6= 0 price setters do pay attention to markup shocks, nonfundamental consumption variance is positive. Hence, in the case gˆ0 ≥ 0, the unique optimal monetary policy is g0 = 0. At this policy, price setters pay no attention to markup shocks and the equilibrium allocation equals the efficient allocation. Next, consider the case gˆ0 < 0 and g¯0 ≤ 0. When gˆ0 = g¯0 there always exists a unique g0 , g¯0 ]. We assume that the equilibrium, while when gˆ0 < g¯0 there exist two equilibria for all g0 ∈ (ˆ central bank evaluates policies that yield multiple equilibria by looking at the worst equilibrium. g0 , g¯0 ] by looking at the equilibrium with This implies that the central bank evaluates policies g0 ∈ (ˆ κ > 0, because the equilibrium with κ > 0 has (i) weakly larger price dispersion, (ii) strictly larger non-fundamental consumption variance, and (iii) strictly larger fundamental consumption variance (due to the fact that the price level increases after a positive markup shock) than the equilibrium i h with κ = 0. Let us turn to optimal monetary policy. Let us first rank policies g0 ∈ − φφλ , gˆ0 . c i h For all g0 ∈ − φφλ , gˆ0 , the unique equilibrium is κ = 0, implying that price dispersion equals c

zero, non-fundamental consumption variance equals zero, and fundamental consumption variance

equals g02 σ 2λ . Since gˆ0 < 0, the central bank’s loss function is a strictly decreasing function of g0 on h i − φφλ , gˆ0 . Next let us rank policies g0 > gˆ0 . In an equilibrium with κ > 0, the derivative of price c

dispersion with respect to g0 is non-negative and the derivative of non-fundamental consumption

variance with respect to g0 is positive. See steps 3-5. Furthermore, in an equilibrium with κ > 0,

28

we have lim (g0 − θλ ) = gˆ0 < 0.

(38)

g0 ↓ˆ g0

See equations (24) and (31). Moreover, if condition (37) holds for all κ > 0, the derivative of g0 −θλ with respect to g0 is non-positive and thus the derivative of fundamental consumption variance with respect to g0 is non-negative. See step 6. Hence, if condition (37) holds for all κ > 0, the central g0 , ∞). Finally, the central bank’s bank’s loss function is a strictly increasing function of g0 on (ˆ loss function is a continuous function of g0 at g0 = gˆ0 . See equations (31), (35), (36), and (38). Hence, if condition (37) holds for all κ > 0, then in the case gˆ0 < 0 and g¯0 ≤ 0, the unique optimal monetary policy is g0 = gˆ0 . At this policy, price setters pay no attention to markup shocks and thus the price level does not respond to markup shocks. Finally, consider the case gˆ0 < 0 and g¯0 > 0. In this case, there is a small complication. The interval (ˆ g0 , g¯0 ] now contains positive values. For these values g0 ∈ (0, g¯0 ], it is unclear whether the equilibrium with κ = 0 or the equilibrium with κ > 0 is worse. On the one hand, price dispersion is weakly larger and non-fundamental consumption variance is strictly larger in the equilibrium with κ > 0. On the other hand, fundamental consumption variance is strictly smaller in the equilibrium with κ > 0, because when g0 > 0 nominal spending increases after a positive markup shock and the fact that the price level increases after a positive markup shock reduces the consumption response. However, this complication can easily be avoided by noticing that policies g0 > 0 cannot be optimal. The reason is simple. Focusing on the equilibrium with κ = 0, the central bank’s objective is a strictly increasing function of g0 on [0, g¯0 ], because price dispersion equals zero, non-fundamental consumption variance equals zero, and fundamental consumption variance equals g02 σ 2λ . Focusing on the equilibrium with κ > 0, the central bank’s objective is again a strictly increasing function of g0 on [0, ∞) so long as condition (37) holds for all κ > 0, because the derivatives of price dispersion and fundamental consumption variance with respect to g0 are non-negative and the derivative of non-fundamental consumption variance with respect to g0 is positive. Hence, if condition (37) holds h i for all κ > 0, a policy g0 > 0 cannot be optimal. We can restrict attention to policies g0 ∈ − φφλ , 0 . c

The same arguments as in the previous paragraph (after replacing the value g¯0 by zero) then yield

the following conclusion. If condition (37) holds for all κ > 0, the unique optimal monetary policy is g0 = gˆ0 . At this policy, price setters pay no attention to markup shocks and thus the price level does not respond to markup shocks. 29

Step 8: Alternative assumption about the central bank. In step 7, we assumed that the central bank evaluates policies that yield multiple equilibria by looking at the worst equilibrium. We now assume instead that the central bank only considers policies that yield a unique equilibrium. If condition (37) holds for all κ > 0, the unique optimal monetary policy is once again ⎧ ⎨ gˆ0 if gˆ0 < 0 . g0∗ = ⎩ 0 otherwise

i h To see this, note the following. Recall that there are three regions: (i) if g0 ∈ − φφλ , gˆ0 there exists c

a unique equilibrium and κ = 0, (ii) if g0 ∈ (ˆ g0 , g¯0 ] there exist two equilibria, one with κ = 0 and

another with κ > 0, and (iii) if g0 > g¯0 there exists a unique equilibrium and κ > 0. Now consider the two cases gˆ0 ≥ 0, and gˆ0 < 0. First, in the case gˆ0 ≥ 0, the exact same arguments as in step 7 yield that the unique optimal monetary policy is g0 = 0. Second, in the case gˆ0 < 0, recall that the h i central bank’s objective is a strictly decreasing function of g0 on − φφλ , gˆ0 . Furthermore, recall c

that when we focus on the equilibrium with κ > 0 for all g0 > gˆ0 , the central bank’s objective is a g0 , ∞) so long as condition (37) holds for all κ > 0, and the strictly increasing function of g0 on (ˆ central bank’s objective is a continuous function of g0 at g0 = gˆ0 . Hence, in the case gˆ0 < 0, we arrive at the following result. If condition (37) holds for all κ > 0, the unique optimal monetary i h policy among all policies satisfying g0 ∈ − φφλ , gˆ0 or g0 > g¯0 is g0 = gˆ0 . c

30

E

More general shocks

This section contains the proof of an analytical result stated in Section 6.4 of the paper. We show that Proposition 4 in the paper extends from the desired markup λt to the more general variable zt introduced in equations (50)-(51) in the paper if the variable zt satisfies the condition − φφz < ϑ. c

The proof follows closely the proof of Proposition 4 in the paper. It consists of the same seven steps. Note that the desired markup λt is a special case of the variable zt with ϑ = 0. The fact that ϑ is now allowed to be different from zero only matters for steps 2 and 7. Hence, apart from notation, only steps 2 and 7 had to be adjusted. This is also where we use the condition − φφz < ϑ. c

Step 1: Equilibrium for a given policy. When the variable zt is i.i.d. over time (ρz = 0) and the central bank responds contemporaneously to z shocks (mt = g0 zt ), the set of rational expectations equilibria of the form pt = θzt for a given monetary policy g0 ∈ R consists of the pairs (θ, κ) ∈ R × R+ solving equations (43)-(44) in the paper, after φλ and σ 2λ have been replaced by φz and σ 2z . Solving equation (43) in the paper for θ yields ¡ ¢ 1 − 2−2κ (φ g0 + φz ) . θ= 1 − (1 − φc ) (1 − 2−2κ ) c

(39)

Furthermore, the first-order condition for the attention problem in equation (44) in the paper reads ω [(1 − φc ) θ + φc g0 + φz ]2 σ 2z ln (2) ≤ 22κ f 0 (κ) with equality if κ > 0.

(40)

Substituting equation (39) into equation (40) and rearranging yields £ ¤2 ω (φc g0 + φz )2 σ 2z ln (2) ≤ f 0 (κ) φc 2κ + (1 − φc ) 2−κ with equality if κ > 0.

(41)

Hence, the pairs (θ, κ) ∈ R × R+ solving equations (43)-(44) in the paper are given by the set of κ ∈ R+ satisfying condition (41) and the corresponding θ ∈ R given by equation (39). Next, it follows from (23), (25), (27), and (50) in the paper as well as si,t = zt + ζ i,t and µ 2 equations ¶ σ that consumption, the profit-maximizing price of good i, and the actual price κ = 12 log2 σz|t−1 2 z|t

of good i for a given (g0 , θ, κ) ∈ R × R × R+ equal ct = (g0 − θ) zt

p∗i,t = [(1 − φc ) θ + φc g0 + φz ] zt ¡ ¢¡ ¢ pi,t = [(1 − φc ) θ + φc g0 + φz ] 1 − 2−2κ zt + ζ i,t , 31

(42) (43)

where σ 2z = 22κ − 1. σ 2ζ

(44)

Step 2: Optimal monetary policy has to satisfy g0 ≥ − φφz . At the monetary policy c

g0 = − φφz , the profit-maximizing price does not respond to z shocks. The unique equilibrium is then c i h i ³ ´2 h (θ, κ) = (0, 0), implying E (pi,t − pt )2 = 0 and E (ct − c∗t )2 = − φφz − ϑ σ 2z . Next, consider c

a monetary policy g0 <

− φφz . c

Price dispersion at the policy g0 <

− φφz c

is weakly larger than price

dispersion at the policy g0 = − φφz because price dispersion is non-negative. Furthermore, inefficient c

consumption variance at the policy g0 < − φφz is strictly larger than inefficient consumption variance c

at the policy g0 =

− φφz c

because for all g0 < − φφz we have c

¡ ¢ 1 − 2−2κ φ (φc g0 + φz ) − ϑ < − z − ϑ < 0. g0 − −2κ 1 − (1 − φc ) (1 − 2 ) φc

Hence, a monetary policy g0 < − φφz cannot be optimal. This means a monetary policy that makes c

the profit-maximizing price fall after a positive z shock cannot be optimal. See equations (39) and (42). From now on, we focus on policies g0 ≥ − φφz . c

Step 3: Equilibrium attention as a function of policy. Let ϕ (κ) denote the right-hand side of the weak inequality in condition (41), that is, ¤2 £ ϕ (κ) = f 0 (κ) φc 2κ + (1 − φc ) 2−κ .

(45)

If ϕ (κ) is a strictly increasing function on R+ , there exists a unique equilibrium for each monetary policy g0 ∈ R. See Proposition 3 in the paper. The unique equilibrium allocation of attention is ⎧ 2 2 0 ⎨ κ f oc if ω (φc g0 + φz ) σ z ln (2) > f (0) κ= , (46) ⎩ 0 otherwise where κf oc is defined implicitly by

ω (φc g0 + φz )2 σ 2z ln (2) = ϕ (κf oc ) .

(47)

If policy induces price setters to pay attention to z shocks (that is, ω (φc g0 + φz )2 σ 2z ln (2) > f 0 (0) and thus κ > 0), then κ = κf oc and the implicit function theorem as well as ϕ0 (κ) 6= 0 yields ∂κ ω2 (φc g0 + φz ) φc σ 2z ln (2) . = ∂g0 ϕ0 (κ) 32

(48)

Hence, for any monetary policy g0 ≥ − φφz that induces price setters to pay attention to z shocks, c

we have

∂κ ∂g0

> 0.

Step 4: Equilibrium price dispersion as a function of policy. If price setters pay no attention to z shocks (κ = 0), price dispersion equals zero. If price setters do pay attention to z shocks (κ > 0), price dispersion is positive. More precisely, when price setters pay attention to z shocks, it follows from equations (43)-(44) as well as equation (24) in the paper that the relative price of good i equals ¡ ¢ pi,t − pt = [(1 − φc ) θ + φc g0 + φz ] 1 − 2−2κ ζ i,t ,

and price dispersion equals

i h ¡ ¢2 E (pi,t − pt )2 = [(1 − φc ) θ + φc g0 + φz ]2 1 − 2−2κ

σ 2z . 22κ − 1

Furthermore, when price setters pay attention to z shocks, it follows from equation (40) that equilibrium attention is given by ω [(1 − φc ) θ + φc g0 + φz ]2 σ 2z ln (2) = 22κ f 0 (κ) . Substituting the last equation into the previous equation yields i f 0 (κ) ¡1 − 2−2κ ¢ h 2 . E (pi,t − pt ) = ω ln (2) Recall that for any monetary policy g0 ≥ − φφz that induces price setters to pay attention to z c

shocks, we have

∂κ ∂g0

> 0. See Step 3. Hence, for any monetary policy g0 ≥ − φφz that induces price c

setters to pay attention to z shocks, the first derivative of equilibrium price dispersion with respect to g0 is positive. Step 5: Equilibrium consumption as a function of policy. Equilibrium consumption equals ct = (g0 − θ) zt . Thus, how the response of consumption to z shocks varies with policy depends on how the response of the price level to z shocks varies with policy. Step 6: Equilibrium price level as a function of policy. The price level equals pt = θzt with θ given by equation (39). If policy induces price setters to pay no attention to z shocks (κ = 0), 33

then θ = 0. By contrast, if policy induces price setters to pay attention to z shocks (κ > 0), then θ > 0 and equation (39) implies ¡ ¢ 1 − 2−2κ ∂κ 2−2κ 2 ln (2) ∂θ φ, = (φc g0 + φz ) + ∂g0 1 − (1 − φc ) (1 − 2−2κ ) c [1 − (1 − φc ) (1 − 2−2κ )]2 ∂g0 where

∂κ ∂g0

is given by equation (48). The first term is the attention effect that is only present in the

model with an endogenous information structure. The second term is the usual effect which is also present in the model with an exogenous information structure. The second term is smaller than one and converges to one as κ goes to infinity. Substituting equation (48) into the last equation yields ¡ ¢ 1 − 2−2κ ω2φc σ 2z ln (2) 2−2κ 2 ln (2) ∂θ 2 (φc g0 + φz ) + φ. = ∂g0 ϕ0 (κ) 1 − (1 − φc ) (1 − 2−2κ ) c [1 − (1 − φc ) (1 − 2−2κ )]2 Furthermore, substituting equations (45)-(47) into the last equation yields ¡ ¢ 1 − 2−2κ ∂θ 4 ln (2) φc f 0 (κ) + φ. = ∂g0 ϕ0 (κ) 1 − (1 − φc ) (1 − 2−2κ ) c Rearranging the right-hand side of the last equation yields ∂θ ∂g0

= = =

2−2κ 4 ln (2) φc f 0 (κ) − +1 ϕ0 (κ) 1 − (1 − φc ) (1 − 2−2κ ) 1 4 ln (2) φc f 0 (κ) − +1 0 2κ ϕ (κ) φc 2 + 1 − φc £ ¤ 4 ln (2) φc f 0 (κ) φc 22κ + 1 − φc − ϕ0 (κ) + 1. ϕ0 (κ) [φc 22κ + 1 − φc ]

The definition of the function ϕ (κ) implies h i £ ¤2 ϕ0 (κ) = f 00 (κ) φc 2κ + (1 − φc ) 2−κ + f 0 (κ) 2 φ2c 22κ − (1 − φc )2 2−2κ ln (2) .

Substituting the last equation into the numerator of the previous equation and rearranging yields £ ¤ φ + (1 − φc ) 2−2κ ∂θ + 1. = f 0 (κ) 2 ln (2) − f 00 (κ) c ∂g0 ϕ0 (κ)

Hence, if policy induces price setters to pay attention to z shocks (κ > 0) and ϕ0 (κ) > 0, then ∂θ ∂g0

≥ 1 if and only if

f 00 (κ) ≤ 2 ln (2) . f 0 (κ)

34

(49)

Step 7: Optimal monetary policy. Let us begin with the case ω (φc ϑ + φz )2 σ 2z ln (2) ≤ f 0 (0). In this case, at the policy g0 = ϑ, price setters pay no attention to z shocks, implying that price dispersion equals zero and inefficient consumption variance equals zero because (g0 − θ − ϑ)2 σ 2z = θ2 σ 2z = 0. Thus, the policy g0 = ϑ achieves the efficient allocation. In addition, any policy g0 6= ϑ does not achieve the efficient allocation. If at the policy g0 6= ϑ price setters pay no attention to z shocks, inefficient consumption variance is positive because (g0 − θ − ϑ)2 σ 2z = (g0 − ϑ)2 σ 2z > 0. If at the policy g0 6= ϑ price setters do pay attention to z

shocks, price dispersion is positive. Hence, in the case ω (φc ϑ + φz )2 σ 2z ln (2) ≤ f 0 (0), the unique

optimal monetary policy is g0 = ϑ. At this policy, price setters pay no attention to z shocks, the price level does not respond to z shocks, and the equilibrium allocation equals the efficient allocation. Let us turn to the case ω (φc ϑ + φz )2 σ 2z ln (2) > f 0 (0). In this case, at the policy g0 = ϑ, price setters do pay attention to z shocks. Consider first equilibrium attention as a function of policy. Let g¯0 ≥ − φφz denote the policy at which price setters stop paying attention to z shocks, c

that is, ω (φc g¯0 + φz )2 σ 2z ln (2) = f 0 (0). The inequality ω (φc ϑ + φz )2 σ 2z ln (2) > f 0 (0) implies that i h g¯0 < ϑ. For all g0 ∈ − φφz , g¯0 we have κ = 0, while for all g0 > g¯0 we have κ > 0. Now, c i h consider price dispersion as a function of policy. For any policy g0 ∈ − φφz , g¯0 we have κ = 0 c

and thus price dispersion equals zero, while for any policy g0 > g¯0 we have κ > 0 and thus price

dispersion is positive. Next, consider inefficient consumption variance as a function of policy. Note h ´ that g0 − θ − ϑ is a continuous function of g0 on − φφz , ∞ , implying that inefficient consumption h ´ c φz variance is a continuous function of g0 on − φ , ∞ . Furthermore, inefficient consumption variance h c h i i is a strictly decreasing function of g0 on − φφz , g¯0 , because for all g0 ∈ − φφz , g¯0 we have κ = 0, c

c

θ = 0, (g0 − θ − ϑ)2 σ 2z = (g0 − ϑ)2 σ 2z , and g0 ≤ g¯0 < ϑ. Finally, if condition (49) holds for all

g0 , ∞), because κ > 0, then inefficient consumption variance is a non-decreasing function of g0 on (¯ at g0 = g¯0 we have g0 − θ − ϑ = g¯0 − ϑ < 0, and for all g0 > g¯0 we have κ > 0 and

∂(g0 −θ) ∂g0

≤ 0. See

Step 6. Hence, if condition (49) holds for all κ > 0, then in the case ω (φc ϑ + φz )2 σ 2z ln (2) > f 0 (0)

the unique optimal monetary policy is g0 = g¯0 . This policy minimizes both price dispersion and inefficient consumption variance. At this policy, price setters pay no attention to z shocks, the price level does not respond to z shocks, price dispersion equals zero, and inefficient consumption variance equals (¯ g0 − ϑ)2 σ 2z . 35

F

Nominal wage rigidity

In this section we study optimal monetary policy in the case of nominal wage rigidity. The results are summarized in Section 6.5 of the paper.

F.1

Assumptions

We assume that the nominal wage rate is fixed at its value in the non-stochastic steady state (i.e., wt = 0) and households commit to supply any amount of labor at this nominal wage rate. This way of introducing nominal wage rigidity is similar to the way Shimer (2012) introduces real wage rigidity. An information-based micro-foundation for this assumption is that households set the nominal wage rate one period in advance and the state of technology and the desired markup are i.i.d. over time (i.e., ρa = ρλ = 0). The assumption that the nominal wage rate does not move with the current state of the economy changes equation (26) in the paper. The equation for the profit-maximizing price now reads ˜ ct − φ ˜ at + φ λt , p∗i,t = (1 − φw ) pt + φ c a λ

(50)

where φw = ˜ = φ c ˜ = φ a

1 1+ 1+ 1+

,

1−α 1+Λ α Λ 1−α α , 1−α 1+Λ α Λ 1 α . 1−α 1+Λ α Λ

Equation (50) replaces equation (26) in the paper. Otherwise the two Ramsey problems in Section 3.2 of the paper remain unchanged. ˜ = 0, the central bank ˜ > 0. When φ Finally, in this section, we assume α < 1 to ensure φ c c cannot affect outcomes. Furthermore, for some of the following derivations, it will be useful to know that

˜ ˜ φ (1 + ψ) (1 − α) φ φa < a. = a ˜ (1 + ψ) (1 − α) + α (γ + ψ) ˜ φc φ φ c c

The inequality follows from α (γ + ψ) > 0. 36

F.2

Optimal monetary policy response to technology shocks

In this subsection, we study the optimal monetary policy response to technology shocks under different assumptions about the information structure. Throughout this subsection, we assume σ 2a > 0, ρa = 0, and σ 2λ = λ−1 = 0. We consider policies of the form mt = f0 εt and equilibria of the form pt = θa at . Perfect information. Under perfect information, the unique equilibrium for a given policy is ct =

˜ φw φ a mt + a. ˜ ˜c t φw + φc φw + φ

This follows from equation (50), pt = mt − ct , and pt = p∗i,t . The central bank can affect the perfect-information allocation because the price level affects the real wage due to nominal wage rigidity. The unique optimal monetary policy is " Ã !# ˜ ˜ φa φ φ φ a + c − a at . mt = ˜ φc φw φc φ c At this policy, the equilibrium allocation equals the efficient allocation ci,t = ct =

φa at , φc

and the equilibrium price level strictly falls in response to a positive technology shock à ! ˜ ˜ φ φ φ a − a at . pt = c ˜c φw φc φ | {z } <0

The optimal monetary policy makes the price level drop after a positive technology shock to ensure that the real wage increases after a positive technology shock. Complete price stabilization in response to technology shocks is suboptimal. The reason is the nominal wage rigidity. Exogenous information. The next proposition concerns the optimal monetary policy response to technology shocks when there is imperfect information and the information structure is exogenous. Proposition 4 Consider the Ramsey problem (21)-(33) in the paper, where the variances of noise are exogenous. Replace equation (26) in the paper by equation (50). Assume σ 2a > 0, ρa = 0, and σ 2λ = λ−1 = 0. Consider policies of the form mt = f0 εt and equilibria of the form pt = θa at . Then, 37

there exists a unique equilibrium for any policy f0 ∈ R, and if σ 2η > 0, the unique optimal monetary policy is

f0∗ =

Ã

˜ φa φ c ˜ φc φ a

˜ +φ c ³

˜ φa φ c ˜ −1 φc φ a 2 ση +φw σ2 a

σ2η σ2a

!

+ φw

³

´2

σ2η σ2a

+ φw σ2

´2

σ2

2

˜ + δ ση2 φ c a

2

˜ + δ ση2 φ c

˜ φ a . ˜ φ c

a

At the optimal monetary policy, the price level strictly falls in response to a positive technology shock, there is inefficient price dispersion, and composite consumption strictly increases in response to a positive technology shock, increasing by more than the efficient response: ³ ´³ ´ 2 ˜ 1 + σσa2 φw 1 − φφa φφ˜ c c a η ˜ a η i,t , φ pi,t − pt = − ³ 2 ´2 2 2 σ σa η ˜ + φw + δ φc σ 2η σ2a ´³ ´ ³ 2 ˜ 1 + σσa2 φw 1 − φφa φφ˜ c c a η ˜ at , pt = − φ ³ ´2 a 2 σ 2a σ 2η ˜ + δφ 2 2 +φ ση

ct =

σ2a σ 2η

³

σ2η σ2a

σ 2a σ2η

w

σa

³

+ φw

σ 2η σ 2a

´2

+ φw

φa φc ´2

c

˜ 2 φ˜ a + δφ cφ ˜ c

˜2 + δφ c

at .

Proof. Step 1: Substituting ct = mt − pt , mt = f0 at , pt = θa at , and λt = 0 into the equation for the profit-maximizing price (50) yields p∗i,t =

´ i h³ ˜ − φ θa + φ ˜ f0 − φ ˜ at . 1−φ c w c a

The price of good i in period t then equals pi,t =

´ i h³ ˜ c − φw θa + φ ˜ c f0 − φ ˜a 1−φ

and the price level in period t equals pt =

¢ σ 2a ¡ a , + η t i,t σ 2a + σ 2η

h³ ´ i ˜ − φ θa + φ ˜ f0 − φ ˜ 1−φ c w c a

σ 2a at . σ 2a + σ 2η

Thus, the unique rational expectations equilibrium of the form pt = θa at is given by the solution to the following equation θa =

´ i h³ ˜ − φ θa + φ ˜ f0 − φ ˜ 1−φ c w c a 38

σ 2a . σ 2a + σ 2η

Solving the last equation for θa yields ˜ ˜ f0 − φ φ c a

θa =

˜ +φ + φ c w

σ2η σ 2a

.

Hence, for a given monetary policy f0 ∈ R, we have pt =

˜ ˜ f0 − φ φ c a

at ,

(51)

˜a ˜ c f0 − φ φ

η i,t .

(52)

˜ +φ + φ c w

pi,t − pt =

˜ c + φw + φ

σ 2η σ2a σ 2η σ2a

Substituting the monetary policy mt = f0 at and the equilibrium price level into ct = mt − pt yields ´ ³ 2 ση ˜ f0 + φ + φ 2 w a σa ct = at . (53) 2 ˜ + φ + ση2 φ c w σ a

The last three equations characterize the equilibrium for a given monetary policy f0 ∈ R. Step 2: Substituting equations (51)-(53) and λt = 0 into the central bank’s objective yields ⎡⎛ ³

1 ⎢⎜ ⎣⎝ 1−β

σ 2η σ 2a

´ ˜a + φw f0 + φ

˜ c + φw + φ

−

σ2η σ 2a

⎞2

⎛

˜ φc φ a

=

c

³

ση +φw σ2 a σ2η w σ2a

+φ

c

w

σ2a

w

σa

´2

σ2

σa

2

˜ + δ ση2 φ c

c

˜ φ a . ˜c φ

a

Substituting the optimal monetary policy into equations (51)-(53) yields ³ ´³ ´ 2 ˜ 1 + σσa2 φw 1 − φφa φφ˜ c c a η ˜ a at , φ pt = − ³ 2 ´2 2 2 σ σa η ˜ + φw + δ φc σ 2η σ2a ´³ ´ ³ 2 ˜ 1 + σσa2 φw 1 − φφa φφ˜ c c a η ˜ a η i,t , φ pi,t − pt = − ³ ´2 2 σ 2a σ 2η ˜ + δφ 2 2 +φ ση

ct =

σ2a σ 2η

³

σ 2a σ2η

w

σa

σ2η σ2a

⎤

˜ c f0 − φ ˜a φ φa ⎟ 2 ⎠ σ 2η ⎥ ⎠ σa + δ ⎝ ⎦. σ 2η φc ˜ φ +φ +

If σ 2η > 0, the unique f0 ∈ R that minimizes this expression is à ! ˜ φa φ ³ 2 ´2 c −1 ˜ ˜ ση σ2 ˜ 2 φa φc ˜ φc2 φa +φ + φ + δ η2 φ 2 f0∗

⎞2

³

+ φw

σ 2η σ 2a

´2

+ φw

39

φa φc ´2

c

˜ 2 φ˜ a + δφ cφ ˜ c

˜2 + δφ c

at .

At the optimal monetary policy, the consumption level, ct , responds to a technology shock more than the efficient consumption level, c∗t . This follows from

φa φc

<

˜ φ a ˜ . φ c

Endogenous information. We now study the optimal monetary policy response to technology shocks when the information structure is endogenous. Consider the Ramsey problem (21)-(37) in the paper, where signal precisions are endogenous. Replace again equation (26) in the paper by equation (50) because of the nominal wage rigidity. Recall that in the economy with an exogenous information structure, complete price stabilization in response to technology shocks is never optimal. See the previous two paragraphs. By contrast, in the economy with an endogenous information structure, complete price stabilization in response to technology shocks is optimal so long as f 0 (0) is large enough. First, we prove that in the economy with an endogenous information structure, the central bank can even achieve the efficient allocation so long as ¶ µ φa ˜ 2 2 0 ˜ − φa σ a ln (2) . f (0) ≥ ω φc φc The proof is simple. If the central bank conducts the policy f0 =

(54) φa φc

pt = 0), then the profit-maximizing price of good i in period t equals ¶ µ ˜ φa − φ ˜ at . p∗i,t = φ c a φc

and κ = 0 (which implies

(55)

Next turn to the attention problem given by equations (34)-(37) in the paper. When the profitmaximizing price satisfies equation (55), κ = 0 is indeed optimal if and only if condition (54) holds. Hence, at the monetary policy f0 =

φa φc ,

κ = 0 is an equilibrium if and only if condition (54) holds.

Finally, if the central bank conducts the policy f0 =

φa φc

and κ = 0, then pi,t − pt = 0 and ct = c∗t .

Second, when f 0 (0) does not satisfy condition (54), the central bank cannot achieve the efficient allocation. However, complete price stabilization in response to technology shocks is still optimal (despite the nominal wage rigidity) so long as f 0 (0) is not too small. We have solved numerically for the optimal monetary policy response to technology shocks for the linear cost function, f (κ) = μκ. For all parameter configurations we tried, we find that there exists a threshold for μ/ω above which complete price stabilization in response to technology shocks is optimal (despite the nominal wage rigidity) that is well below the one given by condition (54). For example, consider the following parameter values that are standard in the business cycle literature: α = 2/3, γ = ψ = 1, 40

h ³ ´i Λ = 1/4, σ2a = (0.0085)2 / 1 − (0.95)2 .8 For these parameter values, the threshold for μ/ω

above which complete price stabilization in response to technology shocks is optimal (despite the nominal wage rigidity) equals 3 ∗ 10−5 . This threshold for the marginal cost of paying attention is one order of magnitude smaller than the one given by condition (54). Furthermore, this threshold for the marginal cost of paying attention is one order of magnitude smaller than the range used by Ma´ckowiak and Wiederholt (2012) to match U.S. business cycle dynamics.

F.3

Optimal monetary policy response to markup shocks

In this subsection, we study the optimal monetary policy response to markup shocks under different assumptions about the information structure. Throughout the subsection, we assume σ 2λ > 0, ρλ = 0, and σ 2a = a−1 = 0. We consider policies of the form mt = g0 ν t and equilibria of the form pt = θλ λt . Perfect information. Under perfect information, the unique equilibrium for a given policy is ct =

φw φλ mt − λ. ˜ ˜c t φw + φc φw + φ

This follows from equation (50), pt = mt − ct , and pt = p∗i,t . The central bank can affect the perfect-information allocation because the price level affects the real wage due to nominal wage rigidity. The unique optimal monetary policy is mt =

φλ λt . φw

At this policy, the equilibrium allocation equals the efficient allocation ci,t = ct = 0, and the equilibrium price level strictly increases in response to a positive markup shock pt =

φλ λt . φw |{z} >0

The optimal monetary policy makes the price level increase after a positive markup shock to ensure that the real wage falls after a positive markup shock. Complete price stabilization in response to markup shocks is suboptimal. The reason is the nominal wage rigidity. 8

See the discussion of parameters in online Appendix C.2.

41

Exogenous information. The next proposition concerns the optimal monetary policy response to markup shocks when there is exogenous imperfect information. Proposition 5 Consider the Ramsey problem (21)-(33) in the paper, where the variances of noise are exogenous. Replace equation (26) in the paper by equation (50). Assume σ 2λ > 0, ρλ = 0, and σ 2a = a−1 = 0. Consider policies of the form mt = g0 ν t and equilibria of the form pt = θλ λt . Then, there exists a unique equilibrium for any policy g0 ∈ R, and if σ 2ζ > 0, the unique optimal monetary policy is

³ ´ 2 ˜ c σ2ζ + φw 1 − δφ σλ φλ . g0∗ = µ ¶ 2 σ2ζ 2 σ 2ζ ˜ 2 + φw + δ φ cσ σ2 λ

λ

At the optimal monetary policy, the price level strictly increases in response to a positive markup shock, there is inefficient price dispersion, and composite consumption strictly falls in response to a positive markup shock: pi,t − pt =

pt =

µ µ

σ2ζ σ2λ

σ 2ζ σ 2λ

+ φw φλ ζ i,t , ¶2 2 σ2ζ ˜ + φw + δ φc σ2 λ

σ2ζ σ2λ

σ 2ζ σ 2λ

ct = − µ

+ φw φλ λt , ¶2 2 σ2ζ ˜ + φw + δ φc σ2

σ2ζ σ2λ

λ

2 ˜ σ2ζ δφ cσ ¶2λ

+ φw

2

˜ 2 σ2ζ + δφ cσ

φλ λt .

λ

Proof. Step 1: Substituting ct = mt − pt , mt = g0 λt , pt = θλ λt , and at = 0 into the equation for the profit-maximizing price (50) yields ´ i h³ ˜ c − φw θλ + φ ˜ c g0 + φλ λt . p∗i,t = 1 − φ

The price of good i in period t then equals ´ i h³ ˜ − φ θλ + φ ˜ g0 + φ pi,t = 1 − φ c w c λ

¢ σ 2λ ¡ λt + ζ i,t , 2 2 σλ + σζ

and the price level in period t equals ´ i h³ ˜ − φ θλ + φ ˜ g0 + φ pt = 1 − φ c w c λ 42

σ 2λ λt . σ 2λ + σ 2ζ

Thus, the unique rational expectations equilibrium of the form pt = θλ λt is given by the solution to the following equation θλ =

h³ ´ i ˜ − φ θλ + φ ˜ g0 + φ 1−φ c w c λ

σ 2λ . σ 2λ + σ 2ζ

Solving the last equation for θλ yields θλ =

˜ c g0 + φλ φ ˜ c + φw + φ

σ2ζ σ2λ

.

Hence, for a given monetary policy g0 ∈ R, we have pt =

pi,t − pt =

˜ g0 + φ φ c λ ˜ +φ + φ c w

σ2ζ σ2λ

˜ c g0 + φλ φ ˜ c + φw + φ

σ2ζ σ2λ

λt ,

(56)

ζ i,t .

(57)

Substituting the monetary policy mt = g0 λt and the equilibrium price level into ct = mt − pt yields ¶ µ 2 σζ + φ w g0 − φλ σ 2λ λt . (58) ct = 2 ˜ + φ + σ2ζ φ c w σ λ

The last three equations characterize the equilibrium for a given monetary policy g0 ∈ R. Step 2: Substituting equations (57)-(58) and at = 0 into the central bank’s objective yields ⎡⎛ µ ⎤ ¶ ⎞2 ⎛ ⎞2 σ2ζ + φw g0 − φλ ⎟ ⎥ ˜ g0 + φ ⎟ σ 2λ 1 ⎢ ⎢⎜ ⎜ φ c λ 2 2⎥ ⎟ σ + δ σ ⎢⎜ ⎝ ⎠ 2 2 ζ⎥ . ⎠ λ ⎦ 1 − β ⎣⎝ φ ˜ + φ + σ2ζ ˜ + φ + σ2ζ φ c w c w σ σ λ

λ

If σ 2ζ > 0, the unique g0 ∈ R that minimizes this expression is

´ 2 ³ ˜ σ2ζ + φ 1 − δφ c σ w λ φλ . g0∗ = µ ¶ 2 σ2ζ 2 σ 2ζ ˜ + φw + δ φc σ2 σ2 λ

λ

Finally, substituting the optimal monetary policy g0∗ into equations (56)-(58) yields the last three equations of the proposition. Endogenous information. We now study the optimal monetary policy response to markup shocks when the information structure is endogenous. Consider the Ramsey problem (21)-(37) in the 43

paper, where signal precisions are endogenous. Replace again equation (26) in the paper by equation (50) because of the nominal wage rigidity. Recall that in the economy with an exogenous information structure, complete price stabilization in response to markup shocks is never optimal. See the previous paragraphs. By contrast, in the economy with an endogenous information structure, complete price stabilization in response to markup shocks is optimal so long as f 0 (0) is large enough. First, we prove that in the economy with an endogenous information structure, the central bank can even achieve the efficient allocation so long as f 0 (0) ≥ ωφ2λ σ 2λ ln (2) .

(59)

The proof is simple. If the central bank conducts the policy g0 = 0 and κ = 0 (which implies pt = 0), then the profit-maximizing price of good i in period t equals p∗i,t = φλ λt .

(60)

Next turn to the attention problem given by equations (34)-(37) in the paper. When the profitmaximizing price satisfies equation (60), κ = 0 is indeed optimal if and only if condition (59) holds. Hence, at the monetary policy g0 = 0, κ = 0 is an equilibrium if and only if condition (59) holds. Finally, if the central bank conducts the policy g0 = 0 and κ = 0, then pi,t − pt = 0 and ct = 0. Second, when f 0 (0) does not satisfy condition (59), the central bank cannot achieve the efficient allocation. However, complete price stabilization in response to markup shocks is still optimal (despite the nominal wage rigidity) so long as f 0 (0) is not too small. We have solved numerically for the optimal monetary policy response to markup shocks for the linear cost function, f (κ) = μκ. For all parameter configurations we tried, we find that there exists a threshold for μ/ω above which complete price stabilization in response to markup shocks is optimal (despite the nominal wage rigidity) that is well below the one given by condition (59). For example, consider the following parameter values that are standard in the business cycle literature: α = 2/3, γ = ψ = 1, and ´i h ³ Λ = 1/4. Furthermore, set the variance of the desired markup to σ 2λ = (0.05)2 / 1 − (0.9)2 .9

For these parameter values, the threshold for μ/ω above which complete price stabilization in response to markup shocks is optimal (despite the nominal wage rigidity) equals 2 ∗ 10−5 . This 9

See the discussion of parameters in online Appendix C.2.

44

threshold for the marginal cost of paying attention is one order of magnitude smaller than the one given by condition (59). Furthermore, this threshold for the marginal cost of paying attention is one order of magnitude smaller than the range used by Ma´ckowiak and Wiederholt (2012) to match U.S. business cycle dynamics.

45

G

Price setters who also take input decisions

In this section we discuss in detail results presented in Section 6.6 of the paper.

G.1

Assumptions

Angeletos and La’O (2012) show that complete price stabilization in response to technology shocks is suboptimal when price setters have imperfect information and the same individuals who set prices also take input decisions. In their model the information structure is exogenous. We revisit this point in our model with an endogenous information structure. We now assume that the same individual who sets the price of good i also takes an input decision. The representative household now supplies two different types of labor, L1,t and L2,t . The preferences of the representative household are given by "∞ Ã 1+ψ1 1+ψ2 !# 1−γ X L L C − 1 1,t 2,t t − E0 βt − , 1−γ 1 + ψ1 1 + ψ2

(61)

t=0

where ψ 1 ≥ 0 and ψ 2 ≥ 0. The production function of firm i is given by 1 2 Yi,t = At Lα1,i,t Lα2,i,t ,

where α1 > 0 and α2 > 0 with α1 + α2 ≤ 1. Here L1,i,t and L2,i,t are firm i’s labor inputs. We consider two alternative assumptions: (i) the same individual who sets the price of good i also chooses the level of one of the two labor inputs in every period, and (ii) the same individual who sets the price of good i also chooses the labor ratio L1,i,t /L2,i,t in every period. Assumption (i) follows closely Angeletos and La’O (2012). The rest of the economy is modeled as in Section 2. Under assumption (i), the level of the labor input that is not chosen by the price setter adjusts to satisfy incoming demand. Under assumption (ii), the level of both labor inputs adjusts to satisfy incoming demand, with the labor ratio chosen by the price setter.10

G.2

The central bank’s objective

In this subsection, we derive the objective of the central bank. In the following subsections, we state and solve the Ramsey problems. 10

The assumption that firms choose two types of labor inputs is for simplicity. An alternative environment is one

where firms choose investment in physical capital or labor effort in addition to a single labor input. See Angeletos and La’O (2012) for these alternative environments.

46

At a feasible allocation the representative household has to supply the labor that is needed to produce the consumption vector I X

L1,t =

i=1 I X

L2,t =

L1,i,t , L2,i,t .

i=1

The production function can be written as ˆ α1 Lα1 +α2 = At Lα1 +α2 L ˆ −α2 , Yi,t = At L i,t 2,i,t 1,i,t i,t ˆ i,t ≡ (L1,i,t /L2,i,t ) denotes the labor ratio at firm i in period t. Substituting the last where L equation into the previous two equations and using Yi,t = Ci,t yields L1,t

L2,t

! 1 α1 +α2 Ci,t = , ˆ −α2 L A t i,t i=1 Ã ! 1 I α1 +α2 X Ci,t = . ˆ α1 At L I X

Ã

i,t

i=1

Furthermore, equation (2) in the paper can be written as CˆI,t =

Ã

I−

I−1 X

1 1+Λt

Cˆi,t

i=1

!1+Λt

,

where Cˆi,t ≡ (Ci,t /Ct ) denotes relative consumption of good i in period t. Substituting the last three equations into the period utility function in (61) yields the following expression for period utility at a feasible allocation ´ ³ ˆ 1,t , . . . , L ˆ I,t , At , Λt U Ct , Cˆ1,t , . . . , CˆI−1,t , L

=

Ct1−γ − 1 1−γ ⎡ Ã Ã ! 1 ! 1 ⎤1+ψ1 µ ¶ 1+ψ1 X I−1 α1 +α2 ˆ ˆ α +α CI,t α1 +α2 ⎦ Ci,t Ct 1 2 ⎣ 1 − + ˆ −α2 ˆ −α2 1 + ψ1 At L L i,t I,t i=1

1 − 1 + ψ2

µ

Ct At

¶

1+ψ 2 α1 +α2

⎡ Ã Ã ! 1 ! 1 ⎤1+ψ2 I−1 X ˆI,t α1 +α2 ˆi,t α1 +α2 C C ⎣ ⎦ + , ˆ α1 ˆ α1 L L i=1

i,t

47

I,t

(62)

where CˆI,t is given by the previous equation. Hence, period utility at a feasible allocation is a function only of the consumption vector, the labor ratios at the different firms, aggregate productivity, and the desired markup in that period. Computing a log-quadratic approximation of the period utility function (62) around the nonstochastic steady state and substituting in the log-linearized demand function for good i yields the following expression for the expected discounted sum of utility losses due to deviations of the allocation from the efficient allocation: # "∞ ´ ³ X ˆ 1,t , . . . , L ˆ I,t , At , Λt β t U Ct , Cˆ1,t , . . . , CˆI−1,t , L E −E

t=0 "∞ X t=0

# ³ ´ ∗ ∗ ˆ ∗1,t , . . . , L ˆ ∗I,t , At , Λt β t U Ct∗ , Cˆ1,t , . . . , CˆI−1,t ,L ⎡

´ ³ ´2 ³ ∗ )2 + δ (c − c∗ ) ˆ ∗ +δ ˆ ∗ ˆ ˆ δ l (c − c l − l − l c t t t cl t l t t t t 1 C 1−γ ⎢ I I ³ ´ E⎢ ≈ − X X 2 21−β ⎣ ˆli,t − ˆl∗ +δ pi I1 (pi,t − pt )2 + δ li I1 i,t i=1

i=1

where ˆli,t is the labor ratio of firm i in period t, ˆlt =

1 I

I X

⎤

⎥ ⎥, ⎦

(63)

ˆli,t is the average labor ratio in period t,

i=1

α1 (1 + ψ 1 ) + α2 (1 + ψ2 ) (α1 + α2 )2 α1 α2 (α2 ψ 1 + α1 ψ 2 ) (α1 + α2 )2 α1 α2 (ψ 2 − ψ 1 ) −2 (α1 + α2 )2 ¶ ∙ ¸ µ 1 1+Λ 2 1 − Λ α1 + α2 1 + Λ α1 α2 , α1 + α2

δc = γ − 1 +

(64)

δl =

(65)

δ cl = δ pi = δ li =

(66) (67) (68)

and the efficient consumption level and the efficient labor ratios are given by c∗t =

γ

α1 (1+ψ1 )+α2 (1+ψ2 ) (α1 +α2 )2 at 1 )+α2 (1+ψ 2 ) − 1 + α1 (1+ψ 2 (α1 +α2 )

ˆl∗ = ˆl∗ = i,t t

+ γ

α1 α2 (ψ2 −ψ1 ) (α1 +α2 )2 ˆl∗ t 1 )+α2 (1+ψ 2 ) − 1 + α1 (1+ψ 2 (α1 +α2 )

ψ2 − ψ1 (c∗ − at ) . α2 (1 + ψ 1 ) + α1 (1 + ψ 2 ) t

48

(69) (70)

G.3

The Ramsey problem when price setters choose the labor ratio

In this subsection, we state the Ramsey problem for the economy with exogenous information and the Ramsey problem for the economy with endogenous information when price setters choose the labor ratio. Before stating these Ramsey problems, we derive a simple expression for the loss in profit that a firm incurs if the price setter chooses a price and a labor ratio that deviate from the optimal decisions under perfect information. Nominal profit of firm i in period t equals Ã

´ ³ ˆ i,t , Pt , Ct , W1,t , W2,t , At , Λt = (1 + τ p ) Pi,t Ci,t −W1,t Π Pi,t , L where Ci,t =

Ã

Pi,t 1 I Pt

  1+ Λ1

!−

Ci,t ˆ −α2 At L i,t

!

1 α1 +α2

−W2,t

Ã

Ci,t ˆ α1 At L i,t

!

1 α1 +α2

t

Ct .

Thus, real profit of firm i in period t (i.e., nominal profit divided by Pt times the marginal utility of consumption of the representative household) is ´ ³ ˆ i,t , Pt , Ct , W1,t , W2,t , At , Λt π Pi,t , L !− 1 Ã Λt 1 Pi,t −γ Ct = Ct (1 + τ p ) I 1I Pt 1   ⎛³ ⎞ α +α ´− 1+ 1 1 2 Λ P t i,t ∙ α2 −α1 ¸ C t 1 ⎜ ⎟ W2,t ˆ α1 +α2 ⎜ I Pt W1,t ˆ α1 +α2 ⎟ −Ct−γ + . Li,t L ⎝ ⎠ Pt Pt i,t At

Expressing the real profit function π in terms of log-deviations from the non-stochastic steady state yields the following real profit function π ˆ ´ ³ π ˆ pi,t , ˆli,t , pt , ct , w1,t , w2,t , at , λt

1 − 1 (p −p )+(1−γ)ct = (1 + τ p ) C 1−γ e Λeλt i,t t ∙ I ¸ α2 ˆ −α1 ˆ W2 li,t li,t w1,t −pt + α +α w2,t −pt + α +α −γ W1 1 2 1 2 L1,i e L2,i e −C + P P 1 1 +α2

−α

e

 1+

1 Λeλt

  (pi,t −pt )+ α

1 −γ 1 +α2

49

 ct − α

1 at 1 +α2

.

,

After a quadratic approximation of the real profit function π ˆ around the non-stochastic steady state, the profit-maximizing price and the profit-maximizing labor ratio are given by 1−(α1 +α2 ) 1 α1 +α2 α1 +α2 c + t 1 +α2 ) 1+Λ 1 +α2 ) 1+Λ 1 + 1−(α 1 + 1−(α α1 +α2 Λ α1 +α2 Λ 1 Λ α1 +α2 1+Λ at + λ 1−(α1 +α2 ) 1+Λ t 1 +α2 ) 1+Λ + 1−(α 1 + α1 +α2 Λ α1 +α2 Λ

p♦ i,t = pt + −

1

[α1 (w1,t − pt ) + α2 (w2,t − pt )] (71)

ˆl♦ = − (w1,t − w2,t ) , i,t

(72)

and the loss in profit due to deviations of the price and labor ratio from the profit-maximizing choices equals

´2 ω ³ ´2 ω pp ³ ll ˆ ♦ ˆ pi,t − p♦ l + − l , i,t i,t i,t 2 2

where ω pp ω ll

∙

W1 L1,i + = C P ∙ W1 L1,i + = C −γ P −γ

(73)

¸ ¸ 1+Λ ∙ 1 − (α1 + α2 ) 1 + Λ W2 Λ L2,i 1+ P α1 + α2 α1 + α2 Λ ¸ α1 α2 W2 L2,i . P (α1 + α2 )2

(74) (75)

Here we have used the following steady state relationships W1 P L1,i W1 W2 P L1,i + P L2,i W2 P L2,i W1 W2 P L1,i + P L2,i

1 I

W1 P L1,i

C +

W2 P L2,i

= = =

α1 α1 + α2 α2 α1 + α2 1 . α1 + α2

It is now straightforward to state the Ramsey problem. A log-quadratic approximation of the central bank’s objective and a log-linear approximation of the equilibrium conditions around the non-stochastic steady state yields the following Ramsey problem for the economy with an exogenous information structure: ⎡

´ ³ ´2 ³ ∗ )2 + δ (c − c∗ ) ˆ ∗ +δ ˆ ∗ ˆ ˆ δ l (c − c l − l − l t t cl t l ∞ t t t t X ⎢ c t I I ³ ´ βtE ⎢ min ∞ X X 2 ⎣ {Ft (L),Gt (L)}t=0 ˆli,t − ˆl∗ +δ pi 1I (pi,t − pt )2 + δ li 1I t=0 t i=1

subject to

c∗t

= γ

(1+ψ1 )(1+ψ2 ) α2 (1+ψ1 )+α1 (1+ψ2 ) at , (1+ψ1 )(1+ψ2 ) − 1 + α2 (1+ψ 1 )+α1 (1+ψ 2 )

50

i=1

⎤

⎥ ⎥, ⎦

(76)

(77)

ˆl∗ = t

ψ2 − ψ1 (c∗ − at ) , α2 (1 + ψ 1 ) + α1 (1 + ψ 2 ) t

(78)

ct = mt − pt ,

(79)

I

1X pt = pi,t , I i=1 h i |I pi,t = E p♦ i,t i,t ,

(80) (81)

ˆ p♦ i,t = pt + φc ct − φa at + φλ λt + φl lt ,

(82)

I

X ˆli,t , ˆlt = 1 I i=1 i h ˆli,t = E ˆl♦ |Ii,t , i,t

(83) (84)

ˆl♦ = − ψ 1 − ψ 2 (ct − at ) − ψ 1 α2 + ψ 2 α1 ˆlt , i,t α1 + α2 α1 + α2

(85)

Ii,t = Ii,−1 ∪ {si,0 , si,1 , . . . , si,t } , ⎞ ⎛ at + η i,t ⎠, si,t = ⎝ λt + ζ i,t at = ρa at−1 + εt ,

(88)

λt = ρλ λt−1 + ν t ,

(89)

mt = Ft (L) εt + Gt (L) ν t .

(90)

(86) (87)

and

Here φc = φλ =

1−(α1 +α2 ) α ψ +α ψ +γ+ 1 1 2 2 2 α1 +α2 (α1 +α2 ) 1−(α1 +α2 ) 1+Λ 1+ α +α Λ 1 2 Λ 1+Λ 1−(α1 +α2 ) 1+Λ 1+ α +α Λ 1 2

, φa =

,

φl =

α ψ +α ψ 1 + 1 1 2 22 α1 +α2 (α1 +α2 ) 1−(α1 +α2 ) 1+Λ 1+ α +α Λ 2 α1 α2 1 (ψ1 −ψ2 ) (α1 +α2 )2 1−(α1 +α2 ) 1+Λ 1+ α +α Λ 1 2

, (91)

.

The objective (76) is simply expression (63), where the δ’s are given by equations (64)-(68). Equations (77)-(78) for the efficient consumption level and the efficient labor ratio equal equations (69)(70). The equation for the profit-maximizing price (82) follows from substituting the log-linearized equations for equilibrium wages into equation (71). The equation for the profit-maximizing labor ratio (85) follows from substituting the log-linearized equations for equilibrium wages into equation (72). Equations (86)-(90) are the same as in Section 3.2 of the paper.

51

In the economy with an endogenous information structure, signal precisions are endogenous. They are given by the solution to the attention problem of the price setter in firm i. The price setter takes into account that paying more attention improves the price setting decision and the labor ratio decision: min (1/σ2η ,1/σ2ζ )∈R2+

(

Ei,−1

"∞ X t=0

β

t

µ

) # ´2 ω ³ ´2 ¶ ω pp ³ 1 ll ˆ ♦ ♦ pi,t − pi,t + li,t − ˆli,t f (κ) , + 2 2 1−β

subject to

i h , |I pi,t = E p♦ i,t i,t i h ˆli,t = E ˆl♦ |Ii,t , i,t

and 1 κ = log2 2

Ã

σ 2a|t−1 σ 2a|t

!

1 + log2 2

Ã

σ 2λ|t−1 σ 2λ|t

!

,

where ωpp and ωll are given by equations (74)-(75), respectively.

G.4

Ramsey problem when price setters choose the level of one labor input

In this subsection, we state the Ramsey problems for the alternative assumption that the price setter of a firm chooses the price and the level of one of the two labor inputs (not the labor ratio). Without loss of generality, we assume that price setters choose the second labor input, L2,i,t . The other labor input, L1,i,t , then adjusts to satisfy incoming demand. We begin by deriving a simple expression for the loss in profit that a firm incurs if the price setter chooses a price and a labor input that deviate from the optimal decisions under perfect information. Nominal profit of firm i in period t equals Π (Pi,t , L2,i,t , Pt , Ct , W1,t , W2,t , At , Λt ) = (1 + τ p ) Pi,t Ci,t − W1,t where Ci,t =

Ã

Pi,t 1 I Pt

  1+ Λ1

!−

Ã

Ci,t 2 At Lα2,i,t

!

1 α1

− W2,t L2,i,t ,

t

Ct .

Thus, real profit of firm i in period t (i.e., nominal profit divided by Pt times the marginal utility

52

of consumption of the representative household) is π (Pi,t , L2,i,t , Pt , Ct , W1,t , W2,t , At , Λt ) !− 1 Ã Λt 1 Pi,t −γ Ct = Ct (1 + τ p ) I I1 Pt ⎡ ⎤   Ã !− 1+ 1 1 Ã !1 Λt α1 α1 W2,t Ct ⎢ W1,t Pi,t ⎥ −Ct−γ ⎣ + L2,i,t ⎦ . α2 1 Pt A L P P t t 2,i,t I t

Expressing the real profit function π in terms of log-deviations from the non-stochastic steady state yields the following real profit function π ˆ π ˆ (pi,t , l2,i,t , pt , ct , w1,t , w2,t , at , λt ) 1 − 1 (p −p )+(1−γ)ct = (1 + τ p ) C 1−γ e Λeλt i,t t ∙ I ¸     α W2 w1,t −pt − α1 1+ 1λ (pi,t −pt )+ α1 −γ ct − α1 at − α2 l2,i,t −γ W1 w2,t −pt +l2,i,t −γct t 1 1 1 1 Λe L1,i e L2,i e . −C + P P After a quadratic approximation of the real profit function π ˆ around the non-stochastic steady state, the profit-maximizing price and the profit-maximizing labor input of type 2 are given by 1−(α1 +α2 ) 1 α1 +α2 α1 +α2 c + t 1 +α2 ) 1+Λ 1 +α2 ) 1+Λ 1 + 1−(α 1 + 1−(α α1 +α2 Λ α1 +α2 Λ 1 Λ α1 +α2 1+Λ − a + λ 1−(α1 +α2 ) 1+Λ t 1−(α1 +α2 ) 1+Λ t 1 + α1 +α2 1 + α1 +α2 Λ Λ

p♦ i,t = pt +

♦ l2,i,t =

[α1 (w1,t − pt ) + α2 (w2,t − pt )]

1+Λ ³ ´ 1 α1 Λ p♦ , (ct − at ) + (w ˜1,t − w ˜2,t ) − − p t α1 + α2 α1 + α2 α1 + α2 i,t

(92) (93)

and the loss in profit due to deviations of the price and the labor input from the profit-maximizing choices equals ³ ´2 ³ ´³ ´ ω ´2 ωpp ³ l l ♦ ♦ pi,t − p♦ l2,i,t − l2,i,t + 2 2 l2,i,t − l2,i,t + ω pl2 pi,t − p♦ , i,t i,t 2 2

(94)

where ω pp = ω pl2 = ω l2 l2 =

C −γ α1 + α2 C −γ α1 + α2 C −γ α1 + α2

∙

W1 L1,i + P ∙ W1 L1,i + P ∙ W1 L1,i + P

∙ ¸ ¸ 1 − α1 1 + Λ 1+Λ W2 L2,i 1+ P Λ α1 Λ ¸ 1 + Λ α2 W2 L2,i P Λ α1 ¸ W2 α2 (α1 + α2 ) L2,i . P α1 53

(95) (96) (97)

We can now state the Ramsey problem for the economy with an exogenous information structure when price setters choose one labor input. The problem is very similar to the problem (76)-(90). The only difference is that equations (84)-(85) have to be replaced by the following five equations: ∙ ¸ ˆli,t = 1 − 1 + Λ (pi,t − pt ) + ct − at − (α1 + α2 ) l2,i,t , (98) α1 Λ i h ♦ (99) |Ii,t , l2,i,t = E l2,i,t and

♦ = l2,i,t

where

1+Λ ³ ´ 1 α1 Λ p♦ (ct − at ) + (ψ 1 l1,t − ψ 2 l2,t ) − − p t , α1 + α2 α1 + α2 α1 + α2 i,t

l1,t = ˆlt + l2,t ,

(100)

(101)

I

l2,t =

1X l2,i,t . I

(102)

i=1

Equation (98) gives the labor ratio at firm i in period t. This equation follows from the production function and the log-linearized demand function for good i. Equation (99) specifies the labor input chosen by the price setter at firm i. Equation (100) specifies the profit-maximizing labor input. This equation follows from equation (93) and the equilibrium wage equations. Finally, the last two equations give the equilibrium supplies of type one and type two labor. In the economy with an endogenous information structure, signal precisions are endogenous. They are given by the solution to the attention problem of the price setter in firm i. The price setter takes into account that paying more attention improves the price setting decision and the labor input decision: ⎧ ⎡ ⎛ ∞ ⎨ X Ei,−1 ⎣ βt ⎝ min (1/σ2η ,1/σ2ζ )∈R2+ ⎩

ω pp 2

t=0

subject to

and 1 κ = log2 2

⎫ ³ ´2 ³ ´2 ⎞⎤ ωl2 l2 ♦ ⎬ pi,t − p♦ l + − l 2,i,t i,t 2,i,t ⎠⎦ + 1 f (κ) , ³ ´2³ ´ ♦ ⎭ 1−β l2,i,t − l2,i,t +ωpl2 pi,t − p♦ i,t i h pi,t = E p♦ |I i,t i,t , i h ♦ |Ii,t , l2,i,t = E l2,i,t Ã

σ 2a|t−1 σ 2a|t

!

1 + log2 2

Ã

where ωpp , ωpl2 , and ωl2 l2 are given by equations (95)-(97). 54

σ 2λ|t−1 σ 2λ|t

!

,

G.5

Optimal monetary policy response to technology shocks

We now study the optimal monetary policy response to technology shocks in the case of exogenous information and in the case of endogenous information. We begin with two special cases. In these cases, one can show analytically that the assumption that price setters choose the level of a labor input or the labor ratio has no effect on the optimal monetary policy response to technology shocks. Special cases. Consider first the special case γ = 1. When γ = 1, the central bank can achieve the efficient allocation also when price setters choose the labor ratio or the level of one labor input. The reason is the following. It follows from equations (77)-(78) that when γ = 1, the efficient consumption level moves one-for-one with technology, c∗t = at , and the efficient labor ratio is constant, ˆlt∗ = 0. By setting mt = at the central bank can achieve the efficient allocation. At this policy, equation (82) for the profit-maximizing price after using ct = mt − pt reduces to ˆ p♦ i,t = (1 − φc ) pt + φl lt , and equation (85) for the profit-maximizing labor ratio after using ct = mt − pt reduces to ˆl♦ = ψ 1 − ψ 2 pt − ψ 1 α2 + ψ 2 α1 ˆlt . i,t α1 + α2 α1 + α2 Thus, for the setup where price setters choose the labor ratio, pt = ˆlt = 0, ct = at , and pi,t − pt = 0 is an equilibrium. Furthermore, for the setup where price setters choose the level of one labor input, pt = l1,t = l2,t = ˆlt = 0, ct = at , and pi,t − pt = 0 is again an equilibrium. To show this, one has to use equations (100)-(102) and the production function instead of the last equation. Finally, any other policy (i.e., mt 6= at ) does not achieve the efficient allocation. If price setters put no weight on their noisy signals, the consumption level is inefficient because ct = mt 6= at ; and if price setters do put weight on their noisy signals, there is inefficient price dispersion. Hence, when γ = 1, the unique optimal monetary policy response to technology shocks is mt = at , both when price setters choose the labor ratio and when price setters choose the level of one labor input. Complete price stabilization in response to technology shocks is optimal. This argument applies when the information structure is exogenous and when the information structure is endogenous. Next consider the special case ψ 1 = ψ 2 . When ψ 1 = ψ 2 , the Ramsey problem (76)-(91) equals the Ramsey problem (21)-(33) in the paper once ψ and α in the paper are set to the following values: ψ = ψ 1 = ψ 2 and α = α1 + α2 . To see this, note the following. When ψ 1 = ψ 2 , equation 55

(85) reduces to ˆl♦ = − ψ1 α2 + ψ 2 α1 ˆlt . i,t α1 + α2 Then, the unique solution to equations (83)-(85) for any signal precision is ˆlt = ˆli,t = 0. Moreover, substituting ψ 1 = ψ 2 and ˆlt = ˆli,t = 0 into the central bank’s objective (76)-(78) yields ´ ⎡ ⎤ ¡ 1+Λ ¢2 ³ 1 1 ∞ I − X X Λ α +α 1+Λ 1 1 2 β t E ⎣(ct − c∗t )2 + (pi,t − pt )2 ⎦ , min 1+ψ1 I {Ft (L),Gt (L)}∞ γ − 1 + α1 +α2 t=0 t=0 i=1

where

c∗t =

γ

1+ψ1 α1 +α2 1 − 1 + α1+ψ 1 +α2

at .

Thus, the central bank’s objective equals objective (21)-(22) in the paper once we use ψ 1 = ψ 2 and ˆlt = ˆli,t = 0 and we set ψ = ψ 1 = ψ 2 and α = α1 + α2 . Furthermore, equations (79)-(82) and (86)-(91) also equal equations (23)-(33) in the paper once we set ψ = ψ 1 = ψ 2 and α = α1 + α2 . Hence, when ψ 1 = ψ 2 the Ramsey problem (76)-(91) equals the Ramsey problem (21)-(33) in the paper once ψ and α in the paper are set to the following values: ψ = ψ 1 = ψ 2 and α = α1 + α2 . Similarly, when ψ 1 = ψ 2 the firms’ attention problem in Section G.3 equals the firms’ attention problem in the paper once ψ and α in the paper are set to the following values: ψ = ψ 1 = ψ 2 and α = α1 + α2 . Finally, in the paper we only use two properties of ψ and α: (i) ψ ≥ 0, and (ii) α ∈ (0, 1]. These properties hold when ψ = ψ 1 = ψ 2 and α = α1 + α2 . We arrive at the following conclusion. When ψ 1 = ψ 2 , Proposition 1 in the paper extends to the Ramsey problems in Section G.3. Complete price stabilization in response to technology shocks is optimal. Exogenous information. We now study the optimal monetary policy response to technology shocks in the case of an exogenous information structure. In the previous two paragraphs, we studied two special cases: (1) γ = 1, and (2) price setters choose the labor ratio and ψ 1 = ψ2 . We now turn to the other cases. We assume that σ 2a > 0, ρa = 0, and σ 2λ = λ−1 = 0. We consider policies of the form mt = f0 εt and equilibria of the form pt = θa at and ˆlt =

a at .

Let us begin

with the case of price setters choosing the labor ratio. In this case, one can derive the equilibrium for a given monetary policy f0 ∈ R, substitute the equilibrium equations into the central bank’s objective, and then state the first-order condition for optimal monetary policy. From the first-order condition, one can see that the policy that yields complete price stabilization violates the first-order condition when γ 6= 1 and ψ 1 6= ψ 2 . Hence, when price setters choose the labor ratio, γ 6= 1, and 56

ψ 1 6= ψ 2 , complete price stabilization in response to technology shocks is never optimal. Let us turn to the case where price setters choose the level of one labor input. Here we follow the procedure of Angeletos and La’O (2012). They define the constrained efficient allocation as the allocation that maximizes the ex-ante utility of the representative household subject to the constraint that the actions of an agent are contingent at most on the information set of the agent and the constraint that the planner cannot change the information set of agents. One can show that if l2,i,t responds to technology shocks at the constrained efficient allocation, complete price stabilization in response to technology shocks is suboptimal. Hence, when price setters choose the level of one labor input and that labor input responds to technology shocks at the constrained efficient allocation, complete price stabilization in response to technology shocks is never optimal. To save space, these details are not included here and are available from the authors upon request. Endogenous information. We now turn to the case of an endogenous information structure. We solve numerically for the optimal monetary policy response to technology shocks. In particular, we solve for the optimal f0 over a fine grid, where for each f0 we solve numerically the fixed point problem determining the equilibrium level of κ, and evaluate the associated welfare objective. First, consider the Ramsey problem with endogenous information in Section G.3. That is, price setters choose the labor ratio. We assume γ 6= 1 and ψ 1 6= ψ 2 , because we already know that complete price stabilization in response to technology shocks is optimal when γ = 1 or ψ 1 = ψ 2 . We assume a linear cost function for attention, f (κ) = μκ. We obtain qualitatively similar results for the cost function f (κ) = μ22κ , which are available upon request. We assume σ 2a > 0, ρa = 0, and σ 2λ = λ−1 = 0. We consider policies of the form mt = f0 εt and equilibria of the form pt = θa at and ˆlt =

a at .

For all parameter configurations that we have tried, we find that there exists a threshold

for μ/ω pp above which complete price stabilization in response to technology shocks is optimal. The following table reports the threshold for different values of γ and the following values of the other ´i h ³ parameters: ψ 1 = 1, ψ 2 = 2, α1 = α2 = 1/2, Λ = 1/4, and σ2a = (0.0085)2 / 1 − (0.95)2 .11

Complete price stabilization in response to technology shocks is optimal, for instance, for a value of the marginal cost of paying attention that is one order of magnitude smaller than the range used 11

We tried all combinations of ψ 1 ∈ {1, 2} and ψ2 ∈ {1, 2} , and alternative combinations of α1 ∈ (0, 1) and

α2 ∈ (0, 1) .

57

by Ma´ckowiak and Wiederholt (2012) to match U.S. business cycle dynamics.12

Threshold

γ = 1.5

γ = 2.0

γ = 2.5

γ=3

0.4 ∗ 10−6

1.4 ∗ 10−6

2.5 ∗ 10−6

3.7 ∗ 10−6

Table 1: Threshold for μ/ω pp above which complete price stabilization in response to technology shocks is optimal when price setters choose the labor ratio.

Second, consider the Ramsey problem with endogenous information in Section G.4. That is, price setters choose the level of one of the two labor inputs. We assume γ 6= 1 because we already know that complete price stabilization in response to technology shocks is optimal when γ = 1. We assume f (κ) = μκ, and we again obtain qualitatively similar results for the cost function f (κ) = μ22κ , which are available upon request. We assume σ 2a > 0, ρa = 0, and σ 2λ = λ−1 = 0. We consider policies of the form mt = f0 εt and equilibria of the form pt = θa at and ˆlt =

a at .

For all

parameter configurations that we have tried, we find that there exists a threshold for μ/ω pp above which complete price stabilization in response to technology shocks is optimal. The following table reports the threshold for different values of γ and the same parameter values as in the previous paragraph.

Threshold

γ = 1.5

γ = 2.0

γ = 2.5

γ=3

1.9 ∗ 10−6

3.8 ∗ 10−6

6.2 ∗ 10−6

8.7 ∗ 10−6

Table 2: Threshold for μ/ω pp above which complete price stabilization in response to technology shocks is optimal when price setters choose the level of one of the two labor inputs. 12

Ma´ckowiak and Wiederholt (2010) solve a DSGE model with rational inattention and a Taylor rule. They find

that for a value of (μ/ω pp ) between 1 ∗ 10−4 and 2 ∗ 10−4 the model matches various empirical impulse responses of prices to shocks.

58

G.6

Optimal monetary policy response to markup shocks

Finally, we study the optimal monetary policy response to markup shocks in the case of exogenous information and in the case of endogenous information. Again we begin with a special case. In this special case, one can show analytically that the assumption that price setters choose the labor ratio has no effect on the optimal monetary policy response to markup shocks. Special case. Consider again the following special case: price setters choose the labor ratio and ψ 1 = ψ 2 . In the previous section, we already showed that if ψ 1 = ψ 2 then the Ramsey problems (for exogenous and endogenous information) in Section G.3 equal the Ramsey problems (for exogenous and endogenous information) in Section 3 of the paper once the parameters ψ and α in the paper are set to the following values: ψ = ψ 1 = ψ 2 and α = α1 + α2 . Hence, if ψ 1 = ψ 2 , then all propositions in the paper extend to the Ramsey problems in Section G.3. We arrive at the following conclusion. If price setters choose the labor ratio and ψ 1 = ψ 2 , the assumption that price setters choose the labor ratio in addition to the price does not change results concerning the optimality of complete price stabilization. Exogenous information. We now study the optimal monetary policy response to markup shocks in the case of an exogenous information structure. In the previous paragraph, we studied the following special case: price setters choose the labor ratio and ψ 1 = ψ 2 . We now study all other cases. We assume that σ 2λ > 0, ρλ = 0, and σ 2a = a−1 = 0. We consider policies of the form mt = g0 ν t and equilibria of the form pt = θλ λt and ˆlt =

λ λt .

For each of the two setups

(price setters choose the labor ratio and price setters choose the level of one labor input), one can derive the equilibrium for a given monetary policy g0 ∈ R, substitute the equilibrium equations into the central bank’s objective, and then state the first-order condition for optimal monetary policy. From the first-order condition, one can see that complete price stabilization in response to markup shocks is never optimal. To save space, these details are not included here and are available from the authors upon request. Endogenous information. We now turn to economies with an endogenous information structure. We solve numerically for the optimal monetary policy response to markup shocks. In particular, we solve for the optimal g0 over a fine grid, where for each g0 we solve numerically the fixed point problem determining the equilibrium level of κ, and evaluate the associated welfare objective. First, consider the Ramsey problem with an endogenous information structure in Section G.3. That 59

is, price setters choose the labor ratio. We assume a linear cost function for attention, f (κ) = μκ. We obtain qualitatively similar results for the cost function f (κ) = μ22κ , which are available upon request. We assume σ 2λ > 0, ρλ = 0, and σ 2a = a−1 = 0. Furthermore, we assume ψ 1 6= ψ 2 because we already know that complete price stabilization in response to markup shocks is optimal when ψ 1 = ψ 2 . We consider policies of the form mt = g0 ν t and equilibria of the form pt = θλ λt and ˆlt =

λ λt .

For all parameter configurations that we have tried, we find that there exists a thresh-

old for μ/ω pp above which complete price stabilization in response to markup shocks is optimal. The next table reports the threshold for different values of γ and the following values of the other ´i h ³ parameters: ψ 1 = 1, ψ 2 = 2, α1 = α2 = 1/2, Λ = 1/4, and σ 2λ = (0.05)2 / 1 − (0.9)2 .13

Threshold

γ = 1.0

γ = 1.5

γ = 2.0

γ = 2.5

γ=3

1.9 ∗ 10−6

1.5 ∗ 10−6

1.3 ∗ 10−6

1.2 ∗ 10−6

1.1 ∗ 10−6

Table 3: Threshold for μ/ω pp above which complete price stabilization in response to markup shocks is optimal when price setters choose the labor ratio.

Second, consider the Ramsey problem with endogenous information in Section G.4. That is, price setters choose the level of one of the two labor inputs. We assume f (κ) = μκ, and we again obtain qualitatively similar results for the cost function f (κ) = μ22κ , which are available upon request. We assume σ 2λ > 0, ρλ = 0, and σ 2a = a−1 = 0. We consider policies of the form mt = g0 ν t and equilibria of the form pt = θλ λt and ˆlt =

λ λt .

For all parameter configurations that we have

tried, we find that there exists a threshold for μ/ω pp above which complete price stabilization in response to markup shocks is optimal. The following table reports the threshold for different values of γ and the same parameter values as in the previous paragraph.

Threshold

13

γ = 1.0

γ = 1.5

γ = 2.0

γ = 2.5

γ=3

1.2 ∗ 10−5

1.1 ∗ 10−5

1.0 ∗ 10−5

9.8 ∗ 10−6

9.0 ∗ 10−6

We tried all combinations of ψ 1 ∈ {1, 2} and ψ2 ∈ {1, 2} , and alternative combinations of α1 ∈ (0, 1) and

α2 ∈ (0, 1) .

60

Table 4: Threshold for μ/ω pp above which complete price stabilization in response to markup shocks is optimal when price setters choose the level of one of the two labor inputs.

G.7

Summary

When the same individuals who set prices also choose the labor ratio or the level of a labor input, complete price stabilization in response to technology shocks and in response to markup shocks are still optimal in the model with an endogenous information structure so long as the marginal cost of paying attention exceeds a threshold. This threshold is at least one order of magnitude smaller than the values for the marginal cost of paying attention used by Ma´ckowiak and Wiederholt (2012) to match U.S. business cycle dynamics.

61

H

Central bank information and interest rate rules

H.1

Interest rate rules

So far we have assumed that the central bank directly controls nominal spending and commits to a nominal spending rule of the form of equation (9) in the paper. Now we assume that the central bank directly controls the nominal interest rate and commits to an interest rate rule of the form of equation (10) in the paper. This does not change optimal monetary policy. The reason is the following. The set of equilibria that the central bank can implement with a nominal spending rule of the form (9) equals the set of equilibria that the central bank can implement with an interest rate rule of the form (10). To see this, note the following. So far we have not used the log-linearized consumption Euler equation in the Ramsey problem because in the case of a nominal spending rule the consumption Euler equation only determines the equilibrium nominal interest rate and the nominal interest rate was not of interest up to this point. Now take a law of motion of the economy that is an equilibrium law of motion under some nominal spending rule (9). One can then compute the equilibrium law of motion for the nominal interest rate from the consumption Euler equation. The central bank could commit to this law of motion for the nominal interest rate as an interest rate rule. The law of motion of the economy that we started from would then still be an equilibrium law of motion. Similarly, take a law of motion of the economy that is an equilibrium law of motion under some interest rate rule (10). One can then compute the equilibrium law of motion for nominal spending from equation (23). The central bank could commit to this law of motion for nominal spending as a nominal spending rule. The law of motion of the economy that one started from would then still be an equilibrium law of motion. One issue that arises when the central bank commits to an interest rate rule is that the rule might not induce a unique equilibrium. There is an ongoing debate on how one can achieve unique implementation with an interest rate rule. One proposal is to allow the policy rule to differ on and off the equilibrium path. See Atkeson, Chari and Kehoe (2010).

H.2

Central bank information

So far we have assumed that the central bank has perfect information. We now assume that the central bank only receives a noisy signal about the desired markup st = λt +

62

t

(where the

desired markup and the noise follow independent Gaussian white noise processes) and the central bank commits to a rule of the form mt = g0 st . All other assumptions remain unchanged, that is, otherwise the economy is modeled as in Section 2 of the paper. It turns out that the optimal monetary policy is still given by Proposition 4 in the paper if the information by the central bank is not too noisy (i.e., σ 2 > 0 does not exceed a certain threshold). At this policy, price setters pay no attention to markup shocks, the price level does not respond to markup shocks, price dispersion ¡ ¢ equals zero, and consumption variance equals g02 σ 2λ + σ 2 . The proof consists of the following

steps 0-7.

Step 0: Assume as before that there are no aggregate technology shocks (σ 2a = a−1 = 0). Then, the set of rational expectations equilibria of the form ct = mt − pt , mt = g0 st , and pt = θλt for a given monetary policy g0 ∈ R consists again of the pairs (θ, κ∗ ) ∈ R × R+ solving equations (43)-(44) in the paper. Hence, Proposition 3 in the paper still gives the condition for uniqueness of equilibrium. (Note that the price level cannot respond to the noise in the central bank’s signal because we have not changed the assumptions about the information set of firms, i.e., equations (13)-(14) in the paper. If one assumes instead that firms can also pay attention to the central bank’s current noisy signal before setting prices, one can once again prove that the optimal monetary policy is still given by Proposition 4 in the paper so long as the information by the central bank is not too noisy, but the proof is more complicated.) Step 1: Equilibrium for a given policy. Identical to Step 1 in proof of Proposition 4 in the paper, apart from the equation for consumption and the equation for the profit-maximizing price: ct = (g0 − θ) λt + g0

t

p∗i,t = [(1 − φc ) θ + φc g0 + φλ ] λt + φc g0 t . Step 2: Optimal monetary policy has to satisfy g0 ≥ − φφλ . Almost identical to Step 2 in c

proof of Proposition 4 in the paper. At the monetary policy g0 = − φφλ , the profit-maximizing price c

does not respond to markup shocks, price dispersion equals zero, and consumption variance equals ¢ £ ¤ ³ ´2 ¡ 2 σ λ + σ 2 . At any monetary policy g0 < − φφλ , price dispersion is weakly larger and E c2t = φφλ c

c

consumption variance is strictly larger. Hence, a policy g0 < − φφλ cannot be optimal. c

Step 3: Equilibrium attention as a function of policy. Identical to Step 3 in proof of Proposition 4 in the paper. 63

Step 4: Equilibrium price dispersion as a function of policy. Identical to Step 4 in proof of Proposition 4 in the paper. Step 5: Equilibrium consumption as a function of policy. Equilibrium consumption equals ct = (g0 − θ) λt + g0 t . Thus, equilibrium consumption variance equals £ ¤ E c2t = (g0 − θ)2 σ 2λ + g02 σ 2 .

The last equation implies

£ ¤ µ ¶ ∂E c2t ∂θ = 2 (g0 − θ) 1 − σ 2λ + 2g0 σ 2 . ∂g0 ∂g0

(103)

(104)

Step 6: Equilibrium price level as a function of policy. Identical to Step 6 in proof of Proposition 4 in the paper. Step 7: Optimal monetary policy. Let g¯0 ≥ − φφλ denote the policy at which price setters c

stop paying attention to markup shocks. Formally, the policy is given by ω (φc g¯0 + φλ )2 σ 2λ ln (2) = i h f 0 (0). For all g0 ∈ − φφλ , g¯0 we have κ = 0, while for all g0 > g¯0 we have κ > 0. Note that g¯0 ≥ 0 c

if and only if ωφ2λ σ 2λ ln (2) ≤ f 0 (0).

Consider first the case of g¯0 ≥ 0. In this case, at the policy g0 = 0, price setters pay no attention to markup shocks, implying that price dispersion equals zero and consumption variance equals zero £ ¤ because E c2t = (g0 − θ)2 σ 2λ + g02 σ 2 = 0. Hence, in the case of g¯0 ≥ 0, the policy g0 = 0 achieves the efficient allocation. Furthermore, any policy g0 6= 0 does not achieve the efficient allocation.

If at the policy g0 6= 0 price setters pay no attention to markup shocks, consumption variance is ¡ ¢ £ ¤ positive because E c2t = (g0 − θ)2 σ 2λ + g02 σ 2 = g02 σ 2λ + σ 2 > 0. If at the policy g0 6= 0 price

setters pay attention to markup shocks, price dispersion is positive. Hence, in the case of g¯0 ≥ 0,

the unique optimal monetary policy is g0 = 0. At this policy, price setters pay no attention to markup shocks and the equilibrium allocation equals the efficient allocation. Next, consider the case of g¯0 < 0. Let us begin with price dispersion as a function of policy. i h For any policy g0 ∈ − φφλ , g¯0 we have κ = 0 and thus zero price dispersion. By contrast, for c

any policy g0 > g¯0 we have κ > 0 and thus positive price dispersion. Let us turn to consumption variance as a function of policy. Consumption variance has the following three properties. First, 64

h ´ consumption variance is a continuous function of g0 on − φφλ , ∞ . This follows from equation (103) c

and the properties of θ. Second, consumption variance is a strictly decreasing function of g0 on h i i h £ ¤ ¡ ¢ − φφλ , g¯0 . The reason is that for all g0 ∈ − φφλ , g¯0 we have E c2t = g02 σ 2λ + σ 2 and g0 < 0. c

c

g0 , ∞) as long as condition (46) Third, consumption variance is a non-decreasing function of g0 on (¯

in the paper holds with strict inequality on R++ and the central bank’s noise-to-signal ratio does not exceed the following positive threshold µ ¶µ ¶ σ2 θ ∂θ 1− ≤ inf −1 . g0 ∂g0 σ 2λ g0 ∈(¯g0 ,0)

(105)

g0 , 0) then equation (104) implies The last result follows from the following observations. If g0 ∈ (¯ ∂E [c2t ] that ∂g0 ≥ 0 if and only if ¶µ µ ¶ σ2 ∂θ θ ≤ 1− −1 . g0 ∂g0 σ 2λ The first bracket is positive because θ > 0 for all g0 > g¯0 and g0 < 0. The second bracket is positive because condition (46) in the paper holding with strict inequality on R++ implies

∂θ ∂g0

> 1.

Furthermore, if g0 = 0 then equation (104) implies £ ¤ µ ¶ ∂E c2t ∂θ = 2θ − 1 σ 2λ > 0. ∂g0 ∂g0 In addition, if g0 > 0 then equation (104) implies £ ¤ µ ¶ ∂E c2t ∂θ = −2 (g0 − θ) − 1 σ 2λ + 2g0 σ 2 > 0. ∂g0 ∂g0 The first term on the right-hand side is positive because g0 − θ < 0 at g0 = 0 and

∂θ ∂g0

> 1.

The second term on the right-hand side is positive because g0 > 0. Hence, consumption variance g0 , ∞) as long as condition (46) in the paper holds with is a non-decreasing function of g0 on (¯ strict inequality on R++ and the central bank’s noise-to-signal ratio does not exceed the positive threshold (105). Combining results, we reach the following conclusion. If condition (46) in the paper holds with strict inequality on R++ and the central bank’s noise-to-signal ratio does not exceed the positive threshold (105), then in the case of g¯0 < 0 the unique optimal monetary policy is g0 = g¯0 . This policy minimizes both price dispersion and consumption variance. At this policy, price setters pay no attention to markup shocks, the price level does not respond to markup shocks, ¡ ¢ price dispersion equals zero, and consumption variance equals g¯02 σ 2λ + σ 2 . 65

H.3

Central bank information - numerical example

Figure 6 shows the threshold for the noise-to-signal ratio of the central bank’s signal below which the optimal monetary policy is given by Proposition 4 in the paper. We assume a linear cost function for attention, f (κ) = μκ. We depict the threshold as a function of (μ/ω). The other parameters are set to the values presented and discussed in online Appendix C. Recall the following analytical results from the previous subsection. See Step 7. When g¯0 ≥ 0, the monetary policy g0 = 0 achieves the efficient allocation, because at the inaction policy price setters pay no attention to markup shocks. To implement the policy g0 = 0 the central bank needs no information about the current realization of the desired markup. Hence, when g¯0 ≥ 0, the optimal monetary policy is g0 = 0 (as stated by Proposition 4 in the paper) for any noise-to-signal ratio of the central bank’s signal. Next, when g¯0 < 0, the optimal monetary policy is given by Proposition 4 in the paper if the noise-to-signal ratio of the central bank’s signal satisfies condition (105). For the parameter values presented and discussed in online Appendix C, g¯0 = 0 if and only if (μ/ω) = 3.6 ∗ 10−4 . Hence, if (μ/ω) ≥ 3.6 ∗ 10−4 , the optimal monetary policy is given by Proposition 4 in the paper ¢ ¡ for any noise-to-signal ratio σ 2 /σ 2λ . If (μ/ω) < 3.6 ∗ 10−4 , Figure 6 depicts the right-hand side of ¡ ¢ condition (105). The threshold for σ 2 /σ 2λ is increasing in (μ/ω).

66

I

Micro-foundation of the cost function

This section contains additional results that we did not include in Section 6.8 to save space. First, if the function g (κ) is linear, the cost function for attention f (κ) = κ g (κ) is linear. This provides a micro-foundation of the linear cost function for attention that we assume in some numerical examples. See, for example, Section 5.2. Second, one can relax the assumption that managers have linear disutility of labor. If managers have strictly convex disutility of labor, the equilibrium wage rate of managers depends on the labor supply of managers and thus on the amount of information processing in the economy. One can write the equilibrium real wage rate as (¯ κ) Wtm = . m Pt (Ct )−γ Recall that the variable κ is the amount of attention that the price setter in a firm devotes to aggregate conditions. The variable κ ¯ denotes the average κ in the economy and the term

(¯ κ)

captures the dependence of managers’ marginal disutility of labor on the amount of information processing for price setting purposes. The cost function for attention then has two arguments - the amount of attention that the price setter of the firm devotes to aggregate conditions, κ, and the average amount of attention that price setters devote to aggregate conditions, κ ¯ , because it affects the wage rate of managers: f (κ, κ ¯) = κ

(¯ κ) g (κ) .

(106)

Since the labor market for managers is competitive, firms take the wage rate of managers as given and the first-order condition for the firms’ attention problem reads ω [(1 − φc ) θ + φc g0 + φλ ]2 σ 2λ ln (2) ≤ 22κ

∂f (κ, κ ¯) with equality if κ > 0. ∂κ

(107)

There is a unique κ that satisfies this condition for a given κ ¯ . Hence, any equilibrium is a symmetric equilibrium. In a symmetric equilibrium, we have κ ¯ = κ. Define f 0 (κ) as the marginal cost of paying attention when the firm chooses κ and all other firms choose the same κ. Formally, ¯ ∂f (κ, κ ¯ ) ¯¯ 0 = κ (κ) g 0 (κ) . f (κ) ≡ ∂κ ¯κ¯=κ

(108)

Substituting the equilibrium condition κ ¯ = κ into the first-order condition (107) and using definition (108) yields ω [(1 − φc ) θ + φc g0 + φλ ]2 σ 2λ ln (2) ≤ 22κ f 0 (κ) with equality if κ > 0. 67

(109)

Equation (109) equals equation (64) in the proof of Proposition 4 in the paper. Hence, we obtain again the results stated in Propositions 3-5 in the paper. In particular, if the cost function for attention satisfies condition (46) in the paper on R++ , complete price stabilization in response to markup shocks is optimal. Here f 0 (κ) is the marginal cost of paying attention when the firm chooses κ and all other firms choose the same κ, and f 00 (κ) is the derivative of the function f 0 (κ). To understand how assumptions about managers’ marginal disutility of labor affect the expression for f 00 (κ) /f 0 (κ), consider the following example. With linear disutility of labor we have f 0 (κ) = κ g 0 (κ), implying

f 00 (κ) g 00 (κ) = . f 0 (κ) g 0 (κ)

With strictly convex disutility of labor we have f 0 (κ) = κ f 00 (κ) = f 0 (κ)

0 (κ)

(κ)

+

(κ) g 0 (κ), implying

g 00 (κ) . g 0 (κ)

To interpret the right-hand side of the last equation, consider f 00 (κ) /f 0 (κ) at the point κ = 0. The denominator of the first term,

(0), is the marginal disutility of labor of managers when price

setters devote no attention to aggregate conditions. The numerator of the first term,

0 (0),

is the

derivative of managers’ marginal disutility of labor with respect to price setters’ attention when price setters devote no attention to aggregate conditions. Third, one can relax the assumption that there is a (time-invariant and state-invariant) transfer from the government to managerial households keeping the ratio of marginal utilities κ ≡

Ct−γ / (Ctm )−γ constant across monetary policy rules. Without this transfer, the income of managerial households depends on the amount of information processing in the economy. Hence, the ratio κ depends on the amount of attention that price setters devote to aggregate conditions. The cost function for attention then reads f (κ, κ ¯ ) = κ (¯ κ)

g (κ) .

(110)

The cost function for attention (110) is almost identical to the cost function for attention (106). The only difference is that the dependence on κ ¯ arises through κ not

. The same arguments as in

the previous two paragraphs apply. Hence, we obtain once more the results stated in Propositions 3-5 in the paper. In particular, if the cost function for attention satisfies condition (46) in the paper on R++ , complete price stabilization in response to markup shocks is optimal. Here f 0 (κ) is the

68

marginal cost of paying attention when the firm chooses κ and all other firms choose the same κ, and f 00 (κ) is the derivative of the function f 0 (κ). Fourth, one can also relax the assumption that there are complete asset markets. If one assumes that the manager owns the firm, the cost function for attention is simply f (κ) = where

g (κ) ,

is the marginal disutility of labor of the manager and g (κ) are the units of time that

the manager needs to devote to achieve κ units of information processing. One would then weigh profit by the marginal utility of consumption of the manager (instead of the marginal utility of consumption of a worker) before computing the log-quadratic approximation to the profit function. This affects the expression for ω in equation (37) of the paper. However, this has no consequences for our analytical results, because we do not use the expression for ω anywhere in the proofs.

69

References [1] Angeletos, George-Marios, and Jennifer La’O (2012). “Optimal Monetary Policy with Informational Frictions,” Discussion paper, MIT and University of Chicago. [2] Atkeson, Andrew, V.V. Chari, and Patrick Kehoe (2010). “Sophisticated Monetary Policies,” Quarterly Journal of Economics, 125(1), 47-89. [3] Fernald, John (2009). “A Quarterly, Utilization-Adjusted Series on Total Factor Productivity,” Unpublished manuscript, Federal Reserve Bank of San Francisco. [4] Golosov, Mikhail, and Robert E. Lucas, Jr. (2007). “Menu Costs and Phillips Curves,” Journal of Political Economy, 115(2), 171-199. [5] Ma´ckowiak, Bartosz, and Mirko Wiederholt (2009). “Optimal Sticky Prices under Rational Inattention,” American Economic Review, 99(3), 769-803. [6] Ma´ckowiak, Bartosz, and Mirko Wiederholt (2012). “Business Cycle Dynamics under Rational Inattention,” Discussion paper, ECB and Goethe University Frankfurt. [7] Midrigan, Virgiliu (2011). “Menu Costs, Multi-Product Firms, and Aggregate Fluctuations,” Econometrica, 79(4), 1139—1180. [8] Nakamura, Emi, and Jón Steinsson (2008). “Five Facts about Prices: A Reevaluation of Menu Cost Models,” Quarterly Journal of Economics, 123(4), 1415-1464. [9] Rao, C. R. (1973). Linear Statistical Inference and its Applications, New York: Wiley. [10] Shimer, Robert (2012). “Wage Rigidities and Jobless Recoveries,” Discussion paper, University of Chicago. [11] Smets, Frank, and Rafael Wouters (2007). “Shocks and Frictions in U.S. Business Cycles: A Bayesian DSGE Approach,” American Economic Review, 97(3), 586-606. [12] Woodford, Michael (2002). “Imperfect Common Knowledge and the Effects of Monetary Policy,” In Knowledge, Information, and Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps, ed. Philippe Aghion et al. Princeton and Oxford: Princeton University Press. 70

Figure 1: Set of equilibria at different policies

ϕ (κ )

D

LHS ( g 0′′) LHS ( g 0′ ) LHS ( g 0 )

H

G

LHS ( g 0′′′)

E

B

F

C

A

κ

Figure 2: Set of equilibria at different policies

ϕ (κ )

G

LHS ( g 0′′′) LHS ( g 0 ) LHS ( g 0′′) LHS ( gˆ 0 )

H

A D

B

E

F

C

κ

1

ρ =0

ρ = 0.9

Figure 3: Optimal Monetary Policy, Exogenous Information Structure

m

0.1

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Exogenous Imperfect Information Calvo

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Figure 4: Optimal Monetary Policy, Endogenous Information Structure, Persistent Desired Markup, Signal on h

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ï0.5 ï1 ï1.5 ï2

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Figure 5: Optimal Monetary Policy, Endogenous Information Structure, Persistent Desired Markup, Signal on p*

0

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ï0.5 ï1 ï1.5 ï2

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Figure 6: Threshold for central bank noiseïtoïsignal ratio 1

0.9

Threshold for σ !2 /σ

2 λ

0.8

0.7

0.6

0.5

0.4

0.3

0.2

Complete price level stabilization optimal for any noiseïtoïsignal ratio

0.1

0

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µ /ω

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x 10