CUESTIONARIO DE RELACIÓN Bartholomew y Horowitz,

Under opacity (σ2

φ→∞), firms do not observe the monetary instrument.

They are however aware that the central bank responds to cost-push shocks according to its information and rationally use their private informationui

to infer the monetary instrumentI.

In that case, the second column ofΩ_{in (4.8) consists of zeros as the cen-}

tral bank’s disclosure does not contain any valuable information. The solu- tion to the inference problem of each firm boils down to

Ei(u, I) =E Ã u I ui ! = Ã Ω1 Ω2 ! ui = Ã σ2 u σ2 u+σ2ρ νσ2 u σ2 u+σ2ρ ! ui.

Plugging this into equation (4.7) yields

p = ∞ X k=0 (1−ξ)kh_Ωk+1 1 (1 +ξν)u i = Ω1(1 +ξν) 1−(1−ξ)Ω1 u= σ 2 u σ2 ρ+ξσu2 (1 +ξν)u=γ1u. (4.12)

The optimal monetary policy consists of choosing the instrument (4.6) that minimizes the unconditional expected loss (4.11) subject to the price rule (4.12). The variance of the price level is simply given by

var(p) =γ₁2σ2u,

while the variance of the output gap is

var(c) = (ν−γ1)2σu2+ν2σµ2.

The fixed-point solution to this optimization problem yields the follow- ing equilibrium price setting for firmi:

pi =γ1ui = λσ2 u ξσ2 u+σρ2 · σ 2 uσ4ρ+ξ2σu4σµ2+ 2ξσu2σρ2σµ2+σρ4σ2µ+ξσu4σρ2 ξ2_σ6 u+λσu2σρ4+λξ2σu4σ2µ+ 2λξσu2σρ2σµ2+λσρ4σµ2 ui,

while the optimal monetary policy satisfies

ν = − (ξΩ 2 1−λ(1−Ω1)Ω1)σ2u (ξ2_Ω2 1 +λ(1−Ω1)2)σu2 +λ(1−(1−ξ)Ω1)2σµ2 = λσ 4 uσ2ρ−ξσ6u ξ2_σ6 u+λσu2σρ4+λξ2σu4σµ2 + 2λξσ2uσρ2σ2µ+λσρ4σ2µ . (4.13) Interestingly, under opacity, the optimal monetary policy coefficient (4.13) can be positive or negative depending on the parameter configuration. As discussed above, cost-push shocks create a trade-off between price level and output gap stabilization. The central bank’s disclosure influences the reac- tion of the price level to monetary policy and thereby the trade-off the cen- tral bank faces. Opacity reduces the effectiveness of monetary policy on the price level as it increases fundamental and strategic uncertainty of firms about the central bank’s action. Under opacity, the central bank’s influence on the price level is limited as firms do not observe its instrument. So, con- tracting the aggregate demand is relatively ineffective to reduce the price level. Hence, the central bank may find it optimal to reduce the negative output gap (instead of the price level) by increasing aggregate demand (i.e.

ν >0).

Yet opacity is not a sufficient condition for the policy coefficient to be positive. The sign of the policy coefficient (4.13) depends on the relation

between the degree of strategic complementarities1−ξ, the preference of the central bank for output gap stabilizationλ, and the relative precision of firm’s informationσ2

ρ/σu2. In particular, the following condition holds:

ν >0⇔ξ < λσ 2 ρ σ2 u . (4.14)

We propose to call the case where ν > 0 the blow with the windprinciple, according to which the central bank expands nominal aggregate demand whenever cost-push shocks are positive. We now discuss the conditions for

ν >0.

Degree of strategic complementarities The policy coefficient is pos-

itive when complementarities are high (ξ low). As opacity weakens the effectiveness of monetary policy on the price level, strong complementar- ities reduce it even further. Two related intuitions can be mentioned for this effect to arise. First, when the degree of strategic complementarities in the economy is high, higher-order expectations are given an increasing weight in the price setting. This exacerbates the strategic uncertainty about the monetary instrument that characterizes opacity and reduces the effec- tiveness of monetary policy to stabilize the price level. This renders price level stabilization ineffective compared to output gap stabilization and the central bank finds it optimal to set a monetary policy coefficient that accom- modates the aggregate demand:ν >0. Second, when the degree of strategic complementarities is high, the monetary instrumentIhas a small impact on the price level. The price stabilization is less effective and more difficult to achieve as aggregate demand variations have a smaller impact on the price level. The central bank then faces a trade-off that incites it to stabilize the output gap instead of the price level.

Figure 4.1 computes the central bank’s responseνas a function of strate- gic complementarities1−ξwithσ2

u = 1,σ2ρ= 0.5, andλ= 1for three values

of dispersion of central bank’s signals σ2

µ. As strategic complementarities

increase, strategic uncertainty reduces the effectiveness of monetary policy and the policy coefficient ν increases. Not surprisingly, the strength of the central bank’s response (absolute value ofν) increases with the precision of its signal.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 0 0.5 1 1.5 2 ξ ν σ_µ2 =0 σµ 2 =0.5 σµ2=1

Figure 4.1: Optimal monetary policy under opacity (impact ofxi)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −2.5 −2 −1.5 −1 −0.5 0 0.5 σρ2 ν ξ=0.35 ξ=0.5 ξ=0.65

Figure 4.2: Optimal monetary policy under opacity (impact ofσ2 ρ)

Precision of private information When the relative precision of firms’

private information increases (σ2

ρ/σu2 falls), the fundamental and strategic

uncertainty of firms about the monetary instrument decreases. The reduc- tion of uncertainty makes the monetary policy more effective to stabilize the price level and the trade-off favours the lean against the windprinciple. This increases the incentive of the central bank to reduce price deviation. Firms also respond more strongly to cost-push shocks with more accurate information. This implies that the strength of the central bank’s response increases: the absolute value of the policy coefficient rises.

Figure 4.2 shows the optimal monetary policy as a function of the preci- sion of firms’ private informationσ2

ρ, withσu2 = 1,σµ2 = 0.25, and λ = 1for

three values of complementarities ξ. When the precision of firms’ private information increases (σ2

is reduced and prices respond more strongly to cost-push shocks. This in- creases the variability of the price level and the incentive of the central bank to stabilize the price level.

Central bank’s preference Finally, when the central bank is more in-

clined towards price stabilization, the incentive of the central bank to con- tract the nominal demand in order to reduce the price level increases in a very intuitive way. Then thelean against the wind principle is preferred to theblow with the windprinciple.