Appendix A: Positive analysis

A1: Sector-level inflation

Definitions I first introduce two definitions which will be useful in the proofs tofollow.

Definition 1. The cost-based input-output matrix Ω is anN×N matrix with elementi, j given by the expenditure share on input j in i’s cost:

ωij =pjxijmciyi

Definition 2. The sector-level steady-state labor shares in marginal costs are encodedin the N × 1 vector α with components

αi =wLimciyi

In a steady-state with optimal subsidies it holds that Ω = Ω and α = α.

Proof of Propositions 1 and 2

The proofs of these two propositions rely on the same algebra, therefore I presentthem together.

Our objective is to derive the elasticities of sector-level prices with respect to produc-tivity and the output gap. To do this, we first solve for the change in marginal costsas a function of the change in prices, wages and productivity. We will then solve forthe endogenous response of prices and wages to productivity shocks and the outputgap.

The change in marginal costs is given by:

d logmci = αid logw +∑j

ωijd log pj − d logAi

We can then write the change in sectoral prices as function of the change in marginalcosts using the Calvo assumption:

d log pi = δid logmci (45)

1

so thatd logmci = αid logw − d logAi +

∑j

ωijδjd logmcj

This allows to solve for the change in marginal cost as a function of the change inwages and productivity:

d logmc =(I − Ω∆

)−1

(αd logw − d logA) (46)

The change in consumer prices is

d logP = βTd log p = βT∆d logmc = βT∆(I − Ω∆

)−1

(αd logw − d logA) (47)

From the consumption-leisure trade-off we have

d logw = d logP + (ϕd logL+ γd log y) =

=(ϕd logL+ γy + γynat + d logP

)=

=((γ + ϕ) y + λTd logA+ d logP

)We can then use (47) to solve for the change in wages as a function of the output gapand productivity shocks. We have:

d logw − d logP =

(1− βT∆

(I − Ω∆

)−1

α

)d logw + βT∆

(I − Ω∆

)−1

d logA =

= (γ + ϕ) y + λTd logA

so that

d logw =

(γ + ϕ) y + βT[(I − Ω

)−1

−∆(I − Ω∆

)−1]d logA

1− βT∆(I − Ω∆

)−1

α(48)

Lemma 6 in the supplemental materials shows that the denominator in (48) is alwayswell defined.

To find marginal costs as function of the output gap and productivity shocks, we plug

2

plug (48) into (46):

d logmc =(γ + ϕ)

(I − Ω∆

)−1

α

1− βT∆(I − Ω∆

)−1

αy+

(I − Ω∆

)−1

α[λT − βT∆

(I − Ω∆

)−1]

1− βT∆(I − Ω∆

)−1

α− I

d logA

From the Calvo assumption (45), the price response is

π = (γ + ϕ)∆(I − Ω∆

)−1

α

1− βT∆(I − Ω∆

)−1

αy+

∆(I − Ω∆

)−1

α[λT − βT∆

(I − Ω∆

)−1]

1− βT∆(I − Ω∆

)−1

α− I

d logA (49)

The expressions for the elasticities B and V in Section 3.2 follow immediately from(49).

Proof of Lemma 2:

In our setup labor is the only factor of production. Therefore labor and input sharesmust sum to one:

α + Ω1 = 1

so that (I − Ω)−1 α = 1. The result

∃i, j such that ωijδj < ωij =⇒ (I − Ω∆)−1 α < 1

follows immediately from the fact that each term in the geometric sum

(I − Ω∆)−1 α =(I + Ω∆ + (Ω∆)2 + ...

)α

3

has at least one component that is smaller than in the corresponding term of

(I − Ω)−1 α =(I + Ω + Ω2 + ...

)α

It then follows that

δw =∑i

βiδi[(I − Ω∆)−1 α

]i<∑i

βiδi ≡ Eβ(δ)

Equation 16 is obtained by differentiating (12).

A2: Output gap and aggregate inflation

Proof of Lemma 3:

From the consumers’ optimal labor supply decision we have:

(logw − logwnat

)−(logP − logP nat

)= γ

(logC − logCnat

)+ ϕ

(logL− logLnat

)From the definition of output gap we have

y = d logC − d logCnat

while Lemma 2 implies that

logL− logLnat = d logC − d logCnat

Therefore we have

γ(logC − logCnat

)+ ϕ

(logL− logLnat

)= (γ + ϕ) y

so that (logw − logwnat

)−(logP − logP nat

)= (γ + ϕ) y (50)

We next need to compute the left hand side of (50), which corresponds to the changein real wages induced by markup distortions. To solve for real wages as a functionof sector-level markups we first consider how nominal wages w impact marginal costs

4

and prices. We have:

d logmci = αid logw +∑j

ωijd log pj − d logAi

andd log pi = d logmci + d log µi (51)

⇒ d logmc =(I − Ω

)−1 (αd logw − d logA+ Ωd log µ

)(52)

⇒ d logP = βT (d logmc+ d log µ) = d logw + λT (d log µ− d logA)

It follows thatd logw − d logP = λT (d logA− d log µ) (53)

In the natural outcome the productivity change is the same as in the economy withsticky prices, while markups are constant (d log µ = 0). Therefore we have

(logw − logwnat

)−(logP − logP nat

)= −λTd log µ (54)

Equations (50) and (54) together give the result.

Appendix B: Optimal policy

B1: Welfare function

Proof of Lemma 5:

From equation (52) we have

d logmc =(I − Ω

)−1 (αd logw − d logA+ Ωd log µ

)so that

d log p = d logw +(I − Ω

)−1

(d log µ− d logA)

Therefore for each sector i we have

(d log pi − d log pnati

)−(d logw − d logwnat

)= (I − Ω)−1 d log µ

5

We can then use the pricing equation (2) to substitute for markups as a function ofinflation rates.

Proof of Proposition 4:

In what follows, I will use the second-order approximation

Z − Z∗

Z' log

(Z

Z∗

)+

1

2log

(Z

Z∗

)2

I denote by

z = log

(Z

Z∗

)I will prove below that, to the second-order, the log changein output with respect tothe efficient equilibrium is given by

y = l − d

where d is a second order term.

Using this result we can approximate the utility function around the efficient outcomeas

U − U∗

UcC' y +

1

2y2 +

1

2

UccC

Ucy2 +

UlL

UcC

(l +

1

2

UllN

Ull2)

=

= y +1− γ

2y2 −

(l +

1 + ϕ

2l2)

=

= y +1− γ

2y2 −

(y + d+

1 + ϕ

2y2

)=

= −γ + ϕ

2y2 − d

where the last equality follows from the fact that, to the second order, y2 = y2 andd2 = yd = 0.

I will now derive the approximation

y = l − d

and the explicit expression for the second order component d.

6

Lemma 2 proves that d log y = d logL to a first order. Therefore we have

y = l︸︷︷︸first order

− d︸︷︷︸second order

+higher order terms

Intuitively, the second order term is a productivity loss induced by markup distortions.These markup distortions endogenously arise from productivity shocks when pricesare sticky, and have two effects. First, the relative price of different firms within thesame sector is distorted with respect to the efficient equilibrium, therefore sector-levelproductivities are lower (i.e. more labor is required to produce one unit of sectoraloutput). I will denote the productivity loss from within-sector price distortions bythe vector a, with components

ai ≡ log

(Yi

F (xij, Li)

)− logAi

wherexij =

∫xij(t)dt

Li =

∫Li(t)dt

and Ai is the TFP of sector i. Second, sector-level markups are also distorted, sothat the relative price indexes of different sectors are different from the efficient equi-librium. Cross-sector price distortions result in lower aggregate productivity.

I define sector-level markups as

µi =pimci

where pi is the sectoral price index (note that the marginal cost is the same for allproducers in sector i). I derive a first-order approximation of the “within-sector” andthe “cross-sector” component of the productivity loss, and then compute the secondorder approximation around the efficient steady-state.

Note that aggregate productivity YLcan be expressed as a function of real wages and

labor shares. Denoting the aggregate labor share by Λ = wLGDP

= wLPY

, by definition

7

we can write aggregate output as

Y =1

Λ

w

PL

In log-deviations from steady-state we have:

Y = w − P − Λ + l (55)

The first order change in real wages d logw−d logP is derived in the proof of Lemma 3

(see equation (54)). Combining (54) with (55) we obtain the first-order approximation

d log Y − d logL = λT (a− d log µ)− d log Λ (56)

We then need to compute d log Λ as function of the change in sectoral markups andproductivities.

The consumers’ budget constraint is

PC = wL+ Π− T

where Π are aggregate profits and T is a lump-sum tax used to finance input subsidies.Dividing both sides by PC we find

1 = Λ +Π− TPY

= Λ + λT(

1− 1

µ

)where µ is the vector of sector-level markups defined above. Therefore we have

d log Λ = − 1

Λ

(∑i

dλi

(1− 1

µi

)+∑i

λid log µiµi

)

Using (56) we find that, around the efficient steady state (where µi = 1 ∀i)

d log Y − d logL = λTa︸︷︷︸within sector

+

(λT

Λ− λT

)d log µ︸ ︷︷ ︸

cross-sector

(57)

8

As λT

Λ− λT = 0 around µ = 1, the first-order productivity loss from cross-sector

misallocation is zero. To compute the second-order loss we need to take the secondderivative of the cross-sector component in equation (57) .

Note that, since the first order effect on both cross-sector misallocation and sector-level productivities is zero, the second-order terms in (d logA) (d log µ) are also goingto be zero. Therefore we only need to derive the cross-sector component with respectto sector-level markups. We have:

D2

((λT

Λ− λT

)d log µ

)=

=1

Λ

−(∑i

λid log µiµi

)2

+ 2∑i

dλid log µiµi

+∑i

λiµi

(d log µi)2

−∑i

dλid log µi =

= −1

2

∑i

∑j

d2ijd log µid log µj (58)

where

d2ij =

∑h

∑k

βhβkσhk[(I − Ω)−1

hi − (I − Ω)−1ki

] [(I − Ω)−1

hj − (I − Ω)−1kj

]+

+∑t

λt∑h

∑k

ωthωtkθthk

[(I − Ω)−1

hi − (I − Ω)−1ki

] [(I − Ω)−1

hj − (I − Ω)−1kj

]+

+∑t

λtαt∑h

ωthθthL (I − Ω)−1

hi (I − Ω)−1hj =

= ΦC

((I − Ω)−1

(i) , (I − Ω)−1(j)

)+∑t

λtΦt

((I − Ω)−1

(i) , (I − Ω)−1(j)

)(59)

To derive the welfare loss as a function of sector-level inflation rates we need tosolve for the endogenous change in sector-level markups due to price rigidities. Themapping between the two is given by equation (2):

d log µ = − (I −∆) d logmc = − (I −∆) ∆−1π

9

Therefore we can re-write (58) as

d2 log Y − d2 logL = λTa− 1

2πTD2π

withd2ij =

1− δiδi

1− δjδj

d2ij

It remains to compute the “within-sector” component λTa.

Index by t the different varieties of product i and note that, given the CES assumption,sectoral output can be written as

Yi = AiF (xij, Li)p−εii∫p−εiit dt

(60)

wherexij =

∫xij(t)dt

Li =

∫Li(t)dt

as above. Using the definition of a we have

ai = log

(p−εii∫p−εiit dt

)A first order approximation of ai is given by

dai = εi

[∫p−εiit d log pitdt∫

p−εiit dt−∫p1−εiit d log pitdt∫p1−εiit dt

](61)

Given the Calvo assumption, around the efficient steady state we have that∫p−εiit d log pitdt∫

p−εiit dt=

∫p1−εiit d log pitdt∫p1−εiit dt

= δd logmci

so that dai = 0.

Let’s now compute the second-order loss by deriving (61) a second time with respect

10

to d log pit. We find1

d2ai = εi

[∫(log pit − log pi)

2 dt−(∫

(log pit − log pi) dt

)2]

=

= εi1− δiδi

π2i

We can thus express the second-order welfare loss from within-sector misallocation as

1

2πD1π

where

d1ij =

0 if i 6= j

λiεi1−δiδi

if i = j

Appendix C: Examples

Example 4. Optimal policy in the vertical chain

Consider a negative downstream shock in a two-stage vertical chain, as in Example 1.Can monetary policy do better than implementing a zero output gap? Which sectorshall it seek to stabilize?

Recall from Example 1 that, under zero output gap, marginal costs and prices increasedownstream and fall upstream. A positive output gap raises wages, so that marginalcosts and inflation increase everywhere, with a stronger effect on the upstream sector.As a result prices are stabilized in this sector, while inflation increases even moredownstream. I argue that this is optimal, for two reasons. First, distortions are morecostly in the upstream sector. Second, the positive effect on stabilization upstreamfrom a given increase in the output gap is larger than the negative effect downstream(and viceversa).

Upstream distortions are more costly because they trigger both within- and cross-sector misallocation, while downstream distortions only trigger within-sector misal-location. This is because there is only one consumption good, and U only uses labor

1This is the same as in the traditional NK model ( Gali (2008) Ch.4)

11

as an input. Therefore cross-sector misallocation can happen only across D’s inputs(labor and U ’s product). Whenever U ’s price does not fully adjust to reflect changesin wages, D inefficiently substitutes between U and labor. Monetary policy can offsetthis effect by stabilizing wages, and thereby reducing U ’s desired price adjustment.Since wages fall after a negative productivity shock, wage stabilization requires toimplement a positive output gap. This is reflected in the cross-sector component ofthe optimal output gap:

− (γ + ϕ) θDL(1− δD (1− (1− αD) δU))2

(1− αD)αD︸ ︷︷ ︸input shares

(1− δU)2 (1− δD)︸ ︷︷ ︸∝ 1−δU

δUπU

d logAD > 0

The relative marginal effect of monetary policy on inflation downstream relative toupstream is given by 1 − (1− αD) δU < 1. This reflects the fact that marginal costsare more sensitive to the output gap in the upstream sector, so that this sector iseasier to stabilize. The within-sector component of the optimal output gap is

−ε(γ + ϕ) (1− αD) (1− δU) (1− δD)

(1− δD (1− (1− αD) δU))2

δU︸︷︷︸benefit for U

− δD (1− (1− αD) δU)︸ ︷︷ ︸cost for D

d logAD

were δDδU

is the relative cost of within-sector price dispersion inD and U . Implementinga positive output gap reduces overall within-sector misallocation if and only if thebenefit for U (δU) is greater than the loss forD (δD (1− (1− αD) δU)), which is alwaysthe case when adjustment frequencies are the same in the two sectors.

Example 5. Optimal policy in the horizontal economy

Consider the horizontal economy in Example 2. Here the tradeoff between within-and cross-sector misallocation is particularly stark. Intuitively, stabilizing within-sector misallocation would require all firms to charge the same price. Since somefirms cannot adjust, this is achieved only if all prices remain constant. At the sectorlevel however incomplete price adjustment results in a price response that is too smallrelative to the change in productivity. Therefore reducing cross-sector misallocationrequires a larger price adjustment than that observed under zero output gap.

Formally, the marginal gain of increasing the output gap for within-sector misalloca-

12

tion is given byBTD1π = ε (γ + ϕ)Eβ(1−δ)π (62)

while for cross-sector misallocation it is

BTD2π = −σ (γ + ϕ)Eβ(1−δ)π (63)

I denote by Eβ(1−δ) the expectation computed with probability weightsβi (1− δi)∑j βj (1− δj)

i=1,...,N

The within- and cross-sector components are proportional, with opposite signs. Re-ducing within-sector misallocation is optimal if and only if the corresponding substi-tution elasticity is larger than the cross-sector one.

To determine the sign of the optimal output gap one needs to solve for the response ofinflation in (62) and (63) as a function of sectoral productivity shocks. As in Example2 the optimal output gap depends on the covariance between productivity shocks andadjustment frequencies (although with different weights). This covariance capturesthe competing effect of wage and productivity on marginal costs. To see this we canexpress inflation in (62) and (63) as a function of productivity shocks:

Eβ(1−δ)π = Covβ(1−δ) (δ, d logA) =

∑i βi (1− δi) δi (−d logmci)∑

j βj (1− δj)(64)

Even though within- and cross-sector misallocation depend on the same covariance,this happens for different reasons. Within sectors, the marginal welfare loss is pro-portional to

βi (1− δi) δi (−d logmci) (65)

Misallocation is highest when the fraction of adjusting firms is closer to 12, because

price dispersion is maximal. To fix this, monetary policy should bring marginal costchanges closer to zero. Since marginal costs have unit elasticity with respect to theoutput gap, its marginal effect on within-sector misallocation is proportional to (65).This component of the optimal output gap is positive if and only if marginal costs

13

decrease more in sectors with large consumption share and intermediate adjustmentprobabilities.

For cross-sector misallocation instead only the fraction (1− δi) of non-adjusting firmsmatters, because these are the firms whose relative price is distorted with respect toproducers in other sectors. The marginal contribution of sector i to the welfare lossis given by

βi (1− δi) (−d logmci)

Monetary policy should amplify the price adjustment by flexible firms, so that sectoralprices better reflect the change in productivity. In this case the relevant elasticity isnot the marginal cost one, but the price one, which is given by δi. The marginal effecton cross-sector misallocation is then also proportional to (65), but with opposite sign.

Example 6. Optimal policy in the oil economy

Let’s go back to the simple oil economy in Example 3. Here the production structureis a combination of a vertical chain and a horizontal economy, so that the optimalpolicy has common components with both.

In this economy all the cross-sector misallocation comes from inefficient substitutionby consumers across final goods. There is no misallocation across inputs in produc-tion: oil uses only labor as an input, and the oil price is fully flexible, so that it isnever distorted relative to wages. Therefore the cross-sector component of the optimaloutput gap is the same as in the horizontal economy in Example 5, given by

BTD2π = −σ (γ + ϕ)Eβ(1−δ)π (66)

Within-sector misallocation instead is present both at the final goods stage and atthe labor stage. There is no misallocation across oil producers, becaused we assumedthat they have flexible prices. The final good stage is the same as in the horizontaleconomy, and it is given by

(BTD1π

)hor

= ε (γ + ϕ)Eβ(1−δ)π (67)

From the vertical chain example we know that the labor stage (i.e. the most upstreamstage) is stabilized with a positive output gap. With wage rigidities (δL < 1) labordemand is inefficiently low for the workers who cannot lower theor wage. A positive

14

output gap reduces the desired wage cut, thereby shrinking the gap between adjustingand non-adjusting workers. The corresponding component of the optimal output gapis:

(BTD1π

)vert

= ε (γ + ϕ)δLEβ (1− δ)1− δLEβ (δ)

(1− δL (1− Eβ (δ))

1− δLEβ (δ)

)Eβ(1−δ) (ωoil) > 0

While the vertical component has a clear sign, to build intuition for the horizontalcomponent we need to solve for the response of sectoral inflation rates in (67). This isdifferent from the horizontal economy in Example 5 (see Equation (42)). Substitutingfor inflation in Equation (67) as a function of the productivity shock, we obtain:

BTD2π =

= σ (γ + ϕ)δL (1− Eβ (δ))

1− δLEβ (δ)

[Eβ(1−δ) (δωoil)−

δL (1− Eβ (δ))

1− δLEβ (δ)Eβ(1−δ) (δ)Eβ(1−δ) (ωoil)

]d logAoil

(68)

Equation (68) differs from (42) because we introduced sticky wages. First, the ex-pression in (68) is proportional to the wage flexibility δL. When wages are fully rigid(δL = 0) monetary policy has no effect on marginal costs and markups, and the opti-mal output gap is zero. Second, the covariance Covβ(1−δ) (δ, d logA) from Example 5

is replaced with the expression in square brackets in (68). Here sectoral productivityshocks are replaced by oil input shares, which reflect the pass-through of the oil shockinto marginal costs. The first term in the expression reflects the effect of productivityon marginal costs, while the second is the effect of the equilibrium change in wageson marginal costs. In the presence of wage rigidities (δL < 1) the latter is muted. Asa result we have:

Eβ(1−δ) (δωoil)︸ ︷︷ ︸productivity channel

− δL (1− Eβ (δ))

1− δLEβ (δ)︸ ︷︷ ︸<1

Eβ(1−δ) (δ)Eβ(1−δ) (ωoil)︸ ︷︷ ︸wage channel

> Covβ(1−δ) (δ, ωoil) (69)

Even though the mapping between productivity and inflation is different, the intu-ition behind the optimal policy remains the same as in the horizontal economy. Thecross-sector component of the optimal output gap is negative if and only if marginalcosts decreased relatively more in sectors with large consumption shares and/or in-

15

termediate adjustment probabilities.

The within-sector component instead combines insights from both the horizontal econ-omy and the vertical chain. The optimal response to within-sector misallocation inthe final goods stage is proportional to 69, as in the horizontal economy. However wehave an additional term coming from misallocation in the labor sector, which playsthe same role as the upstream sector in the vertical chain. Overall, the within-sectorcomponent of the optimal output gap is given by

BTD1π = ε (γ + ϕ)δLEβ (1− δ)1− δLEβ (δ)

(1− δL (1− Eβ (δ))

1− δLEβ (δ)

)Eβ(1−δ) (ωoil)︸ ︷︷ ︸

vertical chain

−

Eβ(1−δ) (δωoil)−δL (1− Eβ (δ))

1− δLEβ (δ)Eβ(1−δ) (δ)Eβ(1−δ) (ωoil)︸ ︷︷ ︸

horizontal economy

d logAoil (70)

In principle, the sign of this component depends on the “adjusted” covariance in (69).Quantitatively, the calibration in Section 6.4.2 shows that this covariance is positive.Given that the within-sector elasticity is larger than the cross-sector elasticity in ourcalibration, and the vertical chain term is also positive, the optimal output gap ispositive.

Appendix D: Cost-push shocks and Dynamics

D1: Exogenous cost-push shocks

In this section I extend the model presented in the main text to allow for exoge-nous sector-level cost-push shocks, which I model as a change in producers’ desiredmarkups. I denote these changes by the N × 1 vector d log µD. Lemma (4) derivessectoral inflation rates and the Phillips curve.

Lemma 4. The elasticity of sectoral prices with respect to cost-push shocks is givenby (

BλT

γ + ϕ− V

)(71)

16

The “divine coincidence” Phillips curve is

DC = (γ + ϕ) y + λTd log µD (72)

while the consumer-price Phillips curve is

πC = κy + u+ v (73)

where

u =δw − δA1− δw

λTd logA

v =δµ

1− δwλTd log µD

δµ =βT∆ (I − Ω∆)−1 d log µD

λTd log µD

The two Phillips curves expressed in terms of deviations from steady-state output are

DC = (γ + ϕ) y + λT(d log µD − d logA

)(74)

πC = κy +δµλ

Td log µD − δAλTd logA

1− δw(75)

Similar to the baseline model, the central bank faces a worse trade-off after a cost-push shock than after a negative productivity shock of the same size (i.e. d logA =

−d log µD). We can see this from the inequality

u

λTd logA<

v

λTd log µD

This is because the change in firms’ desired price is the same for the two shocks, butafter the cost-push shock natural output hasn’t changed. In other words, inflationis the same after the two shocks for a given deviation of output from steady-state,but the output gap is lower under the cost-push shock. Therefore for a given output

17

gap the cost-push shock generates higher inflation. Correspondingly, an additive“aggregate” cost-push term appears in the divine coincidence Phillips-curve.

Lemma (5) solves for the optimal policy response.

Lemma 5. The optimal monetary policy response to a cost-push shock d log µD im-plements the output gap

y∗CP =

BT

Dinflation-output trade-off︷ ︸︸ ︷

BλT

γ + ϕ−

propagation︷︸︸︷V

−direct effect︷ ︸︸ ︷

D2∆ (I −∆)−1

(γ + ϕ) + BTDB

d log µD (76)

Under the optimal policy the inflation target derived in Proposition 6 takes value

πφ =(λT − BTD2∆ (I −∆)−1) d log µD (77)

Comparing (76) with (39) above, we see that the optimal response to productivityand cost-push shocks has a common component, given by the “propagation” term in(76). Monetary policy seeks to address the relative price distortions that arise as theshock propagates through the input-output network. These are the same regardless ofwhether the inflation response is triggered by fluctuations in productivity or desiredmarkups.

While productivity shocks cause misallocation only through the propagation channel,by definition cost-push shocks directly distort relative markups. The response of mon-etary policy to this direct effect is caputerd by the third term in (76). Implementinga positive output gap is optimal if it raises marginal costs more in the sectors whichfaced a larger increase in their desired markup. Whenever this is the case the policytarget is positive under the optimal policy, as reflected in the second term of (77).

Finally, the first term in equation (76) comes from the fact that monetary policyfaces a “worse” trade-off under the cost-push shock than under the productivity shock,because natural output has not fallen. This is also captured by the first term of thepolicy target (77). The intuition is the same as in the one-sector model: in the faceof a cost-push shock the central bank trades off the output loss with the increase

18

in inflation. Therefore the optimal output gap is lower than after an equally-sizednegative productivity shock, while the output level and inflation should be higher.2

D2: Dynamics - Main results

Consumers

Consumers’ preferences are given by

U =∞∑t=0

ρt(C1−γt

1− γ− L1+ϕ

t

1 + ϕ

)where ρ is the discount factor, and Ct and Lt are defined as in Section ??.

In each period consumers are subject to the budget constraint

PtCt +Bt+1 ≤ wtLt + Πt − Tt + (1 + it)Bt

where wtLt is labor income, Πt are firm profits (rebated lump-sum to households),Tt is a lump-sum transfer (which the government uses to finance input subsidies tofirms), Bt is the quantity of risk-free bonds paying off in period t owned by thehousehold and it are nominal interest rates.

Consumer optimization yields the Euler equation

Uct = ρ(1 + it+1)E[Uct+1

PctPct+1

](78)

where Pct is the consumer price index at time t. Log-linearizing equation (78) andimposing market clearing for final goods we find

yt = E [yt+1]− 1

γ

(it+1 − E

[πct+1

]− log ρ

)(79)

Re-writing equation (79) in gaps yields

yt = E [yt+1]− 1

γ

(it+1 − E

[πct+1

]− rnt+1

)(80)

2However note that in the multi-sector model this channel is potentially counteracted by theresponse to the “direct” effect.

19

where rnt+1 is the natural interest rate, satisfying

rnt+1 = log ρ+ γλTE [logAt+1 − logAt]

Policy instruments

I consider a cashless economy, in which interest rates are the only policy instrument.At each period t the central bank sets the risk-free rate it+1.

Production

Within each period the production technology is as described in Section 2.3. Sectoralproductivity shifters Ait vary across periods.

As in the one-period model, I assume that the government sets input subsidies to offsetthe markup distortions arising from monopolistic competition. Sectoral subsidies areconstant over time, and given by

1− τi =ε∗it − 1

ε∗it

as in Section 2.3.

All producers minimize costs given wages and input prices. At every time t producersin sector i solve

cit(y) = minxijt,Lit wtLit +∑j

pjtxijt s.t. AitFi (Lit, xijt) = y

With constant returns to scale marginal costs are the same for all firms, and all firmsuse inputs in the same proportions. However not all of them can adjust prices, sothat firms within the same sector end up charging different markups.

Sector-level inflation dynamics

The firms who can update their price solve

p∗it = maxpi E

[∑t

SDFt (1− δi)t Yit(pi) (pi − (1− τi)mcit)

](81)

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The optimal reset price is

p∗it =E∑

t

[εitε∗iSDFt (1− δi)t Yit(pi)mcit

]E∑

tεit−1ε∗i−1

[SDFt (1− δi)t Yit(pi)

] (82)

Log-linearizing equation (82) yields the following expression for sector-level inflationrates:

πit =δi (1− ρ(1− δi))

1− δi(d log µDit − log µit

)+ ρEt [πit+1] (83)

where µit is the “sector-level” markup, satisfying:

log µit = log pit − logmcit

and µDit is the desired markup of firms in sector i.

Equilibrium

Equilibrium is defined in a similar way as in section 2.5.

For given sectoral probabilities of price adjustment δi, sectoral productivity shiftersAit and interest rates it for each period t, general equilibrium is given by a vector offirm-level markups µfit, a vector of prices pit, a nominal wage wt, labor supply Lt,a vector of sectoral outputs yit, a matrix of intermediate input quantities xijt, anda vector of final demands cit for each period t such that: a fraction δi of firms ineach sector i can adjust their price in every period; markups are chosen optimallyby adjusting firms (see problem (81)), while they are such that prices stay constantfor the non-adjusting firms; consumers maximize intertemporal utility subject to thebudget constraints; producers in each sector i minimize costs and charge the relevantmarkup; and markets for all goods and labor clear.

The sales-based Phillips curve

Proposition 1 shows that the “divine coincidence” Phillips curve inherits the propertiesof the Phillips curve in the one-sector model: it has constant slope, and does notdepend on the realization of sectoral productivity shocks; the aggregate cost-push

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shock enters as an additive residual.3

Proposition 1. It holds that

DCt ≡ λT(I − ∆

)∆−1πt = ρE (DCt+1) + κyt + λTd log µDt (84)

whereκ = γ + ϕ

and ∆ is a diagonal matrix with elements

∆ii =δi (1− ρ(1− δi))1− ρδi(1− δi)

Response of inflation rates and markups to productivity and monetaryshocks

Proposition (2) characterizes the evolution of sector-level inflation, inflation expecta-tions and markups for given productivity shocks logAt− logAt−1, monetary policy yt,and past markups. Note that, different from the one-sector model and the sales-basedPhillips curve in Section , the vector of sector-level past markups is a state variable.4

Proposition 2. Denote by

M≡

(BλT

γ + ϕ− V

)(I − ∆

)∆−1

The evolution of sectoral markups and inflation rates is given by the following systemof difference equations:(ρEπt+1

log µt

)=

(M−1 −M−1V

−(I − ∆

)∆−1 (I −M−1) −

(I − ∆

)∆−1M−1V

)(πt

log µt−1

)+

3Note that the sales-based Phillips curve in (28) does not depend on past markups. This is aconsequence of Lemma 7 in Appendix B2 (see Supplemental Materials).

4The actual state variables are “relative” past markups Vd logµt−1. Given these, the system isinvariant to the “aggregate” past markup λT d logµt−1.

22

+

−M−1(Byt + V (logAt − logAt−1)

)− ∆

(I − ∆

)−1

d log µDt

−(I − ∆

)∆−1M−1

(Byt + V (logAt − logAt−1)

) (85)

Proof. See Appendix D3 in the Supplemental Materials

Proposition 2 extends the results in Sections 4.1.2 and 4.1.3 to the dynamic setup.We can re-write the first equation in (85) as

πt = Byt + V (logAt − logAt−1 + log µt−1)︸ ︷︷ ︸productivity and past markups

+

+

(BλT

γ + ϕ− V

)(d log µDt +

(I − ∆

)∆−1ρEπt+1

)︸ ︷︷ ︸

cost-push shock

(86)

The first term contains the elasticities of sectoral inflation rates with respect to pro-ductivity and monetary shocks, which are the same as in the static setup (see Proposi-tions 1 and 2). In addition, we now have to account for inherited markup distortions,due to the fact that some producers could not adjust their price in past periods.Lemma 7 in online Appendix B2 shows that in the static setting inflation respondsin the same way to productivity shocks and to intial markups. This happens becauseboth induce the same desired price change under zero output gap. The last term inequation (86) is the response to a “cost-push” shock, which can have an exogenouscomponent (d log µDt ) and an endogenous component coming from expected futureinflation. Inflation expectations act as a “cost-push” shock because they change thedesired amount of price adjustment for given output gaps and productivity. Thecost-push shock implied by inflation expectations is given by:

d log µDE = (I −∆) ∆−1ρEπt+1 (87)

Consumer price Phillips curve

We can aggregate sectoral inflation rates in equation (86) into the CPI-based Phillipscurve, obtaining

πCt = κyt + ρEπCt+1 + ut + vt (88)

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where

ut =δw − δA1− δw

λT (d logAt − d logAt−1) +δw − δµ−1

1− δwλTd log µt−1

vt =δw − δπC1− δw

ρEπCt+1 +δµD

1− δwλTd log µDt

δµ−1 =βT∆ (I − Ω∆)−1 d log µt−1

λTd log µt−1

δπC =βT∆ (I − Ω∆)−1 (I − Ω)Eπt+1

EπCt+1

δµD =βT∆ (I − Ω∆)−1 d log µD

λTd log µD

Equation (88) highlights that past markups and inflation expectations create an en-dogenous cost-push term in the Phillips curve, in the same way as productivity shocks.

Closing the output gap

Lemma 6 shows that the central bank can implement zero output gap in all periods bytargeting it directly in the Taylor rule, as long as the policy rule is “reactive enough”.

Lemma 6. Assume that the productivity shocks follow an AR1 process:

logAt+1 − logAt = η (logAt − logAt−1) + ut+1

with Eut+1 = 0 and η < 1. Then there is a unique path of inflation rates such that theoutput gap is constantly zero. This equilibrium can be implemented with the interestrate rule

it = rnt + βTEπzgt+1︸ ︷︷ ︸nominal rate under zero output gap

+ ζyt (89)

with ζ > 0.

Proof. See Appendix D3 in the Supplemental Materials.

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Optimal policy

Propositions 3 and 4 characterize the dynamic welfare loss and the central bank’spolicy problem.

Proposition 3. Given a path yt, πt, zt+1∞t=0 for the output gap, sectoral inflationrates and markups, the expected second-order welfare loss is given by

∞∑t=0

ρsE[(γ + ϕ) y2

t + πTt D1πt + zTt+1D2zt+1

]where

zt ≡ −(I − ∆

)−1

∆ log µt−1

The optimal policy problem is

minXXX

∞∑t=0

ρsE[(γ + ϕ) y2

t + πTt D1πt + zTt+1D2zt+1

]subject to (85).

Proposition 4. Consider the optimal policy problem without commitment, where thecentral bank solves

minyt,πt,zt+1 (γ + ϕ) y2t + πTt D1πt + zTt+1D2zt+1

subject to (85). The FOCs yield

y∗t = − BTDV(γ + ϕ) + BTDB

[(d logAt − d logAt−1)− d log µt−1 + (I − Ω) ρEπt+1] (90)

Proof. See Appendix D3 in the Supplemental Materials.

In the same spirit as Proposition 2, Proposition 3 decomposes the optimal output gapinto the response to productivity shocks and past markups, which is the same as inthe static setup (see Proposition 5), and the response to inflation expectations, whichhave a similar effect as a “cost-push” shock (see online Appendix D1).5 Lemma 7 and

5To facilitate the comparison, use the equalityM =(BλT

γ+ϕ − V)

(I −∆) ∆−1 = I + V (I − Ω) .

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Lemma 8 below show that the optimal policy can be implemented with a targetingrule, in the same way as in the static setup (see Proposition 6).

Lemma 7. Assume that the productivity shocks follow an AR1 process:

logAt+1 − logAt = η (logAt − logAt−1) + ut+1

with Eut+1 = 0 and η < 1.

Then there is a unique path of inflation rates such that the optimal output gap (90)is implemented in each period.

Proof. See Appendix D3 in the Supplemental Materials.

Lemma 8. Denote by

φT ≡λT(I − ∆

)∆−1 + BTD

γ + ϕ(91)

For ζ > γ the interest rate rule

it+1 =

real rate under opt policy︷ ︸︸ ︷rnt+1 + γ

[Ey∗t+1 − y∗t

]+βTEπ∗t+1︸ ︷︷ ︸

nominal rate under optimal policy

+ζφTt (πt − ρEπt+1)︸ ︷︷ ︸inflation target

(92)

implements the optimal policy (90).

Proof. See Appendix D3 in the Supplemental Materials.

Comparing equation (70) from the static setup with equation (91) we see that theyyield a very similar inflation target. The key difference is that in the dynamic setup thecentral bank should not just target current inflation, but also inflation expectations.The intuition is that the welfare-relevant variable are sector-level markups, and themapping between current inflation and markups depends of inflation expectations.Specifically, from the pricing equation (83) we have d log µ = (I−∆)∆−1(πt−ρEπt+1).

26