ONLINE APPENDIX. The real effects of monetary shocks in sticky price models: a sufficient statistic approach. F. Alvarez, H. Le Bihan, and F.

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1 ONLINE APPENDIX The real effects of moetary shocks i sticky price models: a sufficiet statistic approach F. Alvarez, H. Le Biha, ad F. Lippi A Comparig BBP to CPI data to estimate kurtosis We match a subset of our Frech CPI data with the prices from 3 Frech retailers take from the Billio Price Project BPP dataset, see Cavallo 5. 5 Table offers two comparisos. The first three colums compare the BPP data from large retailers with our CPI data for a similar type of outlet: to this ed we restrict our dataset to CPI price records i hypermarkets, excludig gasolie. The last two colums compare the BPP data from a large retailer specialized i electroics ad appliaces with the CPI data for goods i the category of appliaces ad electroic we use the Coicop omeclature, collected i outlets type hypermarkets, supermarkets, ad large area specialists. Comparig the values of kurtosis from both data sets suggests that Ω/ζ =, see equatio. We ca apply this magitude to the full sample of CPI data, for which o measuremet error-free couterpart like the BPP exists ad the feasible correctio for heterogeeity is oly partial, to obtai a corrected kurtosis. The umber thus obtaied for the kurtosis is ear 4, so it lays i betwee the kurtosis of the Normal ad the Laplace distributio. B Proofs Proof. of Propositio. Let p =. Defie xt pt σ t for t. Usig Ito s lemma we ca verify that the drift of p is σ, ad hece xt is a Martigale. By the optioal samplig theorem x τ, the process stopped at τ, is also a martigale. The 5 We are extremely grateful to Alberto Cavallo for producig these statistics for us. 4

2 Table : Compariso of the CPI vs. the BPP data i Frace CPI category: Hypermarkets Appliaces ad electroic Data source: BPP BPP CPI BPP CPI retailer retailer 5 Hypermarkets retailer 4 Large ret. electr. duratio moths kurtosis Note: The BPP data are documeted i Cavallo 5. Results were commuicated by the author. For CPI data source is INSEE, mothly price records from Frech CPI, data from 3:4 to :4. The subsample i the third colum features the CPI records for the outlet type hypermarkets. The sub-sample i the 5th colum features the CPI records i the category of appliaces ad electroic, as idetified usig the Coicop omeclature, collected i the followig outlets type: hypermarkets, supermarkets, ad large area specialists. Data are stadardized withi each subsample usig Coicop categories. [ [ E xτ p = E [ pτ p σ E τ p = x = ad sice N p i = [ /E τ p ad V ar p i = E [ pτ p / we obtai the desired result. Proof. of Lemma. First, ote that sice two value fuctios differ by a costat, the all their derivatives are idetical. Hece, if the oe for the discout rate ad arrival rate of free adjustmet r +, satisfies value matchig ad smooth pastig, so does the oe for discout rate ad arrival rate of free adjustmet r,,, for the same boudary. Secod, cosider the rage of iactio, subtractig the value fuctio for the problem with parameters r +, from the oe with parameters r,, ad usig that all the derivatives are idetical, oe verifies that if the Bellma equatio holds for the problem with r +,, so it does for the problem with r,. Proof. of Propositio. The first part is straightforward give Lemma ad Propositio 3 i Alvarez ad Lippi 4. The secod part is derived from the followig implicit expressio determiig ȳ see the proof of Propositio 3 i Alvarez Lippi for the derivatio: ψ = [ B r + ȳ σ + r+ ȳ + ȳ + ȳ i= κ i r + i ȳ i σ + ȳ + ȳ r+ + ȳ i= κ i i + r + i ȳ i where κ i = r + i i s=. So we ca rewrite equatio 5 as: ψ = B σ s++s+ [ ξσ, r +,, ȳ. Sice ȳ as ψ the we ca defie the limit: ψ lim ψ ȳ = B [ lim ξσ, r +,, ȳ r + ȳ 5 r+ȳ Simple aalysis ca be used to show that limȳ ξσ, r +,, ȳ = which gives the expressio i the propositio see the techical Appedix J i Alvarez, Le Biha, ad Lippi 43

3 6 for a detailed derivatio. Proof. of Propositio 3. To characterize N p i we write the Kolmogorov backward equatio for the expected time betwee adjustmets T y which solves: T y = + σ T y + y σ T y for y, ȳ ad T ȳ = see the techical Appedix K i Alvarez, Le Biha, ad Lippi 6 for details o the solutio to this equatio. The the expected umber of adjustmets is give by N p i = /T, subject to T <. The solutio of the secod order ODE for T y has a power series represetatio: T y = i= α i y i, for y [, ȳ, with the followig coditios o its coefficiets {α i }: α = α, α σ i+ = α i+ σ +i i, for i ad where < α < / is chose so that α α i for i, lim i+ i α i = ad = i= α i ȳ i. Moreover, T = α is a icreasig fuctio of ȳ sice α solves: = α + α / ȳ [ + σ i ȳ k + + k σ Note that for i : α i = α i / [i! / + i / σ, ad usig the properties of the Γ fuctio i= k= α i = Γ// Γ/ + i / σ i α /. Solvig for α ad usig L /N p i = T = α. Thus l = i= which is equatio 6. Γ i ȳ / i! Γ Γ i ȳ + i σ i! Γ + i σ i= i Proof. of Propositio 4. We first state a lemma about the desity fy. Lemma 3 Let fy;, σ, ȳ be the desity of y [, ȳ i equatio 7 satisfyig the boudary coditios. For ay k > we have: f y;,, ȳ = f y k ;,, σ k k σ k ȳ. Proof. of Lemma 3. Cosider the fuctio fy;, σ, ȳ solvig equatio 7 ad boudary coditios for give, σ, ȳ. Without loss of geerality set σ = σ ad cosider ȳ = ȳ/k ad = k. Notice that by settig C = C k ad C = C k we verify that the boudary coditios hold because C /C = C /C ad that 7 holds which is readily verified by a chage of variable. We ow prove the propositio. Let w p i ;, l, Std p i be the desity fuctio i 44

4 equatio 9. Next we verify equatio. From the first term i equatio 9 otice that l ω p i ; ȳ = s l ω s p i ; s ȳ where the first equality uses the homogeeity of degree - of ω p i ; y see equatio 8. From the secod term i equatio 9 for ȳ l ω p i ; yfy;, ȳ σ, ȳdy = l s ω s p i ; s y s f ys ;, s σ, ȳs dy where the first equality follows from Lemma 3 for k = /s, ad the homogeeity of degree - of ω,. Further we ote ȳ l s ω s p i ; s y s f ys ȳ ;, s σ, ȳs dy = s 3 l ω s p i ; s y f ys ;,, ȳ dy σ where ȳ σ = ȳ σ, so that l is the same across the two ecoomies. Usig z = y s ȳ s 3 l ω s p i ; s y f ys ;, ȳ, ȳ dy = s l ω s p σ i ; z f z;,, ȳ dz. σ where ȳ = s ȳ, which completes the verificatio of equatio. Proof. of Propositio 5. For ay p R with p ȳ, we write mp; ȳ, σ, to emphasize the depedece o ȳ, σ,. A guess ad verify strategy ca be used to show the followig scalig property of the fuctio m: Let k >, the for all p R with p ȳ: mkp; k ȳ, kσ, = k mp; ȳ, σ, ad mp; ȳ, σ k, k = mp; ȳ, σ,. k It is straightforward to verify that this fuctio satisfies the ODE ad boudary coditios for mp see e.g. the oe i the mai text for the = case. Recall the homogeeity of fy stated i Lemma 3. Fially, ote that the desity gp ca be expressed as a fuctio of the desity fy give i equatio 7 ad the desity of the sum of coordiates of a radom variable uiformly distributed o a dimesioal hypersphere of square radius y, as obtaied i Equatio i Alvarez ad Lippi 4. These properties applied to equatio 6 establish the scalig property stated i the propositio. Proof. of Propositio 6. We first otice that for some special cases a simple aalytic proof is available. These cases cocer = or = with l, ; alteratively, they cocer < < ad l = or l =. See Appedix G for details. We ow assume < ad < l < ad prove that M = Kur p i 6 N p i. The proof 45

5 is structured as follows. First we derive a aalytic expressios for Kur p i 6 N p i ad for M. Each expressio is a power series that ivolves oly two parameters: ad l. Verifyig the equality is readily doe umerically to ay arbitrary degree of accuracy. A simple Matlab code for the verificatio, called solvemp.m, is available o our websites. We first show that Kur p i 6 N p i ca be writte as the right had side of equatio 9. This =/Lφ, Kur p i where L is give i is doe i two steps. First otice that Kur p i N p i Propositio 3. The secod step is to derive a aalytic expressio for the Kur p i. We otice that Kur p i = E p4 i τ y = V ar p i = Q = /σ Q σ 4 Lφ, N p i where τ is the stoppig time associated with a price chage, ad where Qy is the expected fourth momet at the time of adjustmet τ coditioal o a curret squared price gap y, i.e. Qy = E p 4 i τ y = y = 3 + E y τ y = y where yτ is the value of the squared price gap at the stoppig time. Notice that for y [, ȳ the fuctio Qy obeys the o.d.e.: 3y Qy = + + Q y σ + Q y σ y with boudary coditio Qȳ = 3ȳ. The solutio of Q has a power series represetatio + which is easily obtaied by matchig coefficiets ad usig the boudary coditios. Usig this power series i the expressio for Kur p i N p i obtaied i the first step gives the expressio o the right had side of equatio 9. See the techical Appedix L i Alvarez, Le Biha, ad Lippi 6 for details o the algebra. Next we derive a expressio for M, which holds for all < ad l : Lemma 4 Let M ; be the area uder the IRF of output ad f ; be the desity of the ivariat distributio for a ecoomy with products ad parameters ȳ,, σ. Let T + y be the expected time util either yt hits ȳ or that util there is a free adjustmet opportuity, whichever happes first, startig at y = y, for a ecoomy with + products ad the same parameters ȳ,, σ. The M ; = ɛ ȳ [ T + y + T +y y fy; dy. 6 46

6 The fuctio T y is characterized i the proof of Propositio 3 where we give a explicit power series represetatio for this fuctio. The proof of Lemma 4 uses a characterizatio of mp i terms of a two dimesioal vector z, y, where z is the sum of the coordiates of p. The fuctio mz, y solves a PDE whose solutio ca be expressed i terms of T y see the techical Appedix M i Alvarez, Le Biha, ad Lippi 6 for details. To compute the right had side of equatio 6, we separately characterize T + y + T +y y ad fy;. Usig the power series represetatio of T + y see proof of Propositio 3 it is immediate to obtai a power series represetatio of T + y+ T +y y. This gives see the techical Appedix N i Alvarez, Le Biha, ad Lippi 6 for details: T + y + T +y y = i= Γ + [ ȳ i i! Γ +i+ σ + i y i σ Γ + ȳ 7 i i= i! Γ +i+ σ For f we use the characterizatio i equatio 7 i term of modified Bessel fuctios of the first ad secod kid see the techical Appedix O i Alvarez, Le Biha, ad Lippi 6 for a step-by-step derivatio. These fuctios have a power series represetatio, which we use to solve for the two ukow costats C, C. This gives: fy = y σ i= β y i, ȳ σ i= β ȳ i, σ i i σ ȳ σ i= β i, ȳ i+/ ȳ σ i= β ȳ i i, σ ȳ σ i i= β i, i= β i, y i σ ȳ σ i i= β i, i= β i, / 8 ȳ ȳ i i+ σ ȳ σ i where the two sequece of coefficiets β are defied i term of the Γ fuctio as β i, i! Γi + / ad β i, i! Γi + / for i =,,,.... The expressio i equatio 8 holds for all real umbers, except whe is a eve atural umber due to a sigularity of the power expasio of the modified Bessel fuctio of the secod kid. Yet the expressio is cotiuous i. Fially, we establish a equivalece to verify equatio 9: Lemma 5 The equality betwee equatio 6 ad the ratio Kur p i /6N p i, as from 47

7 equatio 9, is equivalet to the followig equality j= γ j +j +s s= γ s [ i= ξ i i= ξ i +i+j j = j= γ j + j 9 i= ρ [ i / i++j i= ξ i +i i= i= ρ i i= ξ ρ i i i= ρ i i+, where the sequeces {γ j, ξ j, ρ j } j= are defied as γ j Γ + j ȳ j! Γ + + j σ, ξ j j! Γ j + ȳ +j σ ad ρ j j! Γ j + j ȳ σ. The derivatio of equatio 9 uses equatio 7 ad equatio 8 to compute equatio 6. Verifyig equatio 9 is straightforward sice both sides are simple fuctios of coverget power series, which are arbitrarily well approximated by a fiite sum. As explaied above, for eve values of this expressio should be uderstood as the limit for k or, umerically, as the sum for values of close to k for k N ad k. Proof. of Propositio 7. The idea is to show that for ay ad l we have Kur p i ; µ = Kur p i ; µ, N p i µ = N p i µ, Mδ; µ = M δ; µ 3 for all µ, δ i a eighborhood of,. Note that differetiatig the last expressio with respect to δ, ad evaluatig it at δ = we obtai that M ; µ = M ; µ. Hece we have that Kur p i ;, N p i ad M ; are symmetric fuctios of iflatio aroud µ =. Hece, if they are differetiable, they must have zero derivative with respect to iflatio at zero iflatio. The symmetry i equatio 3 follows from the symmetry o the firm s problem with respect to positive ad egative drift. To establish this symmetry we proceed i two steps. First we aalyze the symmetry of the decisio problem for the firm of Sectio 3.. Secod, we cosider the approximatio to the GE problem for values µ. All the argumets follow a guess ad verify strategy of a simple ature but with heavy otatio. The techical Appedix S i Alvarez, Le Biha, ad Lippi 6 provides the details of the proof. Proof. of Propositio 8. The proof proceeds by verificatio. We aalyze the coditio that esures that every firm with p = y ȳ before the shock will fid that p ιδ ȳ after the shock, where ι is a vector of oes. See the techical Appedix Q i Alvarez, Le Biha, ad Lippi 6 for a detailed derivatio. Proof. of Lemma To prove the lemma we let zt = x 4 t ad yt = x t. We use Ito s 48

8 lemma to obtai: dyt = σt dt + xt σtdw t or yt = t σs ds + t xs σs dw s [ T ad takig expected values: E [yt = E σt dt = T E [σt dt. Likewise dzt = 6 x t σt dt + 4x 3 t σtdw t [ T the E [zt = 6 E σt yt dt = 6 T E [σt yt dt. Now ote that: [ E σt yt dt [ t = E σt σs ds + t xs σs dw s [ ad usig the idepedece of {W t} ad {σt} we have: E [σt t yt = E σt σs ds. The, replacig this expressio ad oticig that KT = E [zt / E [yt, we obtai the desired result. Proof. of Propositio 9. Defie the probability that ut = if u = i as P t i, or: P t i Pr {ut = u = i} for i {, }. These probabilities are give by: P t = θ [ e θ +θ t, P t = θ [ + θ e θ +θ t. θ + θ θ + θ θ We ow use these probabilities to compute two expressios that appear i equatio. see the techical Appedix T i Alvarez, Le Biha, ad Lippi 6 for the details o the derivatio. The first expressio is the expected secod momet vt E [σt which is give by θ [ vt = E σt = σ + σ. 3 θ + θ θ + θ Notice that this expected variace is idepedet of the horizo t. The secod expressio is kt, s E [ σt σs, or the expected fourth momet over a horizo t coditioal o σt. This is give by [ kt, s = σ θ + σ θ + σ σ θ θ θ + θ θ + θ θ + θ e θ +θ t s θ 3 49

9 Usig equatio 3 ad equatio 3 ito equatio gives KT = σ σ θ θ θ +θ [ σ θ θ +θ + σ θ θ +θ [ T θ + θ + e θ +θ T θ + θ T. Without loss of geerality, sice the expressio is homogeeous of degree zero o σ, σ, we ca set σ =. We ca also use θ = /θ +θ. Fially for ay θ we ca let s = θ /θ +θ to obtai equatio. Proof. of Propositio. First we compute the frequecy of adjustmet. Let T i p deote the expected time to hit a barrier coditioal o the state p. backward equatio gives the followig system of ODEs for the expected times: θ T T = + T θ T T = + T σ σ The Kolmogorov which is symmetric T i p = T i p with boudary coditio T i p =. The solutio is T p = θ +θ p p σ θ +σ θ T p = θ +θ p p σ θ +σ θ e χ p +e χ p e χ p +e χ p + σ θ σ σ e χp +e χp σ θ +σ θ + σ θ σ σ e χp +e χp σ θ +σ θ where χ This implies that the average time betwee price adjustmet is give by = T θ + T θ = θ + θ p θ θ σ σ + N a θ + θ σθ + σθ θ + θ σθ + σθ σ θ +σ θ σ σ e χ p + e χ p or, rewritig i terms of the fudametal parameters that pi dow KT, amely θ, s, ξ, ad the implied parameter ˆρ = θ θ = s we have equatio 3. s Now we tur to computig the cumulative output effect. Use the approximatio Mδ δ M = δ ɛ p m pg p + m pg p dp Next we solve for the terms i the equatio. First cosider the ODE that characterizes m i p: θ m m = p + σ θ m m = p + σ m m 5

10 The fuctio must satisfy m i p = m i p ad the boudaries m i p =. The solutio is m p = θ +θ pp p m p = θ +θ pp p + σ 3σ θ +σ θ σ σ e χp e χp θ σ θ +σ θ e χ p e χ p + σ 3σ θ +σ θ σ σ e χp e χp θ σ θ +σ θ e χ p e χ p p p p p where χ σ θ +σ θ σ σ Fially we compute the ivariat distributio of price gaps. Let g i p be the desity for price gaps i state i which must be symmetric aroud p =, zero at the boudary: g i p =. σ g p = θ g p θ g p σ g p = θ g p θ g p For p [ p, p, the shape of the desities is liear triagular, with desity fuctios g p = θ g p = θ p p θ +θ p p p θ +θ p C O the implied cost of price adjustmet I this sectio we give a characterizatio of the model implicatios for the size of the meu cost, i.e. a mappig betwee observable statistics ad the value of ψ/b or ψ we also discuss how to measure B. We cosider two measures for the cost of price adjustmet: the first oe is the cost of a sigle price adjustmet as a fractio of profits: ψ/. Recall that ψ is the cost that a firm must pay if it decides to adjust all prices istataeously i.e. without waitig for a free adjustmet. Measurig this cost as a fractio of profits trasforms these magitudes ito uits that have a ituitive iterpretatio. The secod measure is the average flow cost of price adjustmet give by: N p i ψ l. This cost measures the average amout of resources that the firm pays to adjust prices per period. The latter measure is useful because it relates more directly to what has bee measured i the data by Levy et al. 997; Zbaracki et al. 4, amely the average cost of a price adjustmet. The ext propositio aalyzes the mappig betwee the scaled meu cost ψ/, ad B, l,, N p i ad V ar p i. Propositio Fix the umber of products ad let r. There is a uique triplet σ,, ψ cosistet with ay triplet l [,, V ar p i > ad N p i >. Moreover, 5

11 fixig ay value l, the meu cost ψ ca be writte as: ψ = B V ar p i N p i Ψ, l 33 where Ψ is oly a fuctio of, l. For all the fuctio Ψ, satisfies: lim Ψ, l = l +, lim l Ψ, l/ lim l Ψ, l/ for, ad Ψ, l =, lim Ψ, l l =, 34 l Ψ, l/ lim Ψ, l/ as l. 35 Equatio 33 shows that for ay fixed ad l [, the meu cost ψ is proportioal to the ratio V ar p i /N p i. Secod, equatio 33 shows that the meu cost is proportioal to B, which measures the beefits of closig a price gap. The parameter B is related to the costat demad elasticity faced by firms η see Sectio 3, so that B = ηη /, which ca be writte i terms of the et markup over margial costs m /η so that B = + m/m. 6 The last expressio is useful to calibrate the model usig empirical estimates of the markup such as the oes by Christopoulou ad Vermeule : the estimated markups average aroud 8% for the US maufacturig sector, ad aroud 36% for market services slightly smaller values are obtaied for Frace, see their Table. 7 A similar value for the US, amely a markup rate of about 33%, is used by Nakamura ad Steisso. The left pael of Figure 6 illustrates the comparative static effect of l ad o the implied meu cost, fixig B V ar p i / N p i, i.e. it plots the fuctio Ψ, l. Fixig a value of it ca be see that the meu cost ψ/ is icreasig i l. Ideed equatio 34 shows that as l, the implied meu cost diverges to +. O the other had, for l = ad =, our versio of Golosov-Lucas s model, the meu cost attais its smallest strictly positive value. Fixig l ad movig across lies shows that the implied fixed cost ψ/ is ot mootoe i the umber of products. Ideed, as stated i equatio 34 for a very small share l the values of ψ/ are icreasig i. O the other had, for larger value of the share l, the order of the implied fixed cost is reversed. The model also has clear predictios about the per period say yearly cost of price adjustmets bore by the firms: l N p i ψ/. I spite of the fact that the cost of a sigle deliberate price adjustmet diverges as l, the total yearly cost of adjustmet coverge to 6 Nakamura ad Steisso otice that lower markups higher values of demad elasticity η must imply higher meu costs, as show by equatio 33. Footote 4 i their paper discusses evidece o the markup rates across several microecoomic studies ad macro papers. 7 The evidece for the US services is cosistet with the gross margis, based o accoutig data, reported i the Aual Retail Trade Survey by the US Cesus see 5

12 Figure 6: Implied cost of price adjustmet Cost of oe price adjustmet ψ/ as % of profits Meu cost / profits * = = = = Yearly cost of adjustmet as % of reveues Yearly costs of adjustmet / reveues * = = = = Share of free adjustmets: l Share of free adjustmets: l All ecoomies i the figures feature Std p i =. ad a markup of 5%. For those i the left pael we set N p i =.5. zero cotiuously. This ca be see i the right pael of Figure 6. A simple trasformatio gives the yearly cost of price adjustmets as a fractio of reveues: ln p i ψ/, where the η scalig by η trasforms the uits from fractio of profits ito fractio of reveues. 8 This statistic is useful because it has empirical couterparts, studied e.g. by Levy et al Usig equatio 33 ad the previous defiitio for the markup yields Yearly costs of price adjustmet Yearly reveues = V ar p i m l Ψ, l 36 Figure 6 plots the two cost measures i equatio 33 ad 36 as fuctios of l, for a ecoomy with N p i =.5, Std p i =. ad a markup m 5% i.e. B =. We see this parametrizatio as beig cosistet with the US data o price adjustmets, markups, ad the size distributio of price chages discussed above. The figure illustrates how observatios o the costs of price adjustmets ca be used to parametrize the model. Levy et al. 997 ad Dutta et al. 999 Table IV ad Table 3, respectively documet that for multi-product stores a hadful of supermarket chais ad oe drugstore chai the average cost of price adjustmet is aroud.7 percet of reveues. For a ecoomy with = a reasoable parametrizatio to fit the size-distributio of price chages the right pael of the figure shows that the model reproduces the yearly cost of.7% of reveues whe 8 Sice R = η Π where R is reveues per good ad Π profits per good. 53

13 the fractio of free adjustmets l is aroud 6%. The left pael i the figure idicates that at this level of l the cost of oe price adjustmet is aroud 5% of profits. Proof. of Propositio. To obtai the expressio i equatio 33 we use the characterizatio of l = L ȳ σ, of Propositio 3, it is equivalet to fix a value of φ ȳ σ. We let the optimal decisio rule be ȳ ψ/b, σ, r +, so that we have: ψ ȳ B, σ, r +, σ = φ To be cosistet with V ar p i ad N p i we have, usig Propositio ad l = Lφ, : N p i = /Lφ, ad σ = Lφ, /V ar p i. Thus, after takig r ad usig the expressio above we ca write: ψ ȳ B, N p iv ar p i, l N p i, l V ar p i = L l; Fixig ad l ad computig the total differetial for this expressio with respect to ψ/b, N p i, V ar p i, ad deotig by η ψ, η σ, η the elasticities of ȳ with respect to ψ/b, σ, we have: η ψ ˆψ + ησ ˆN p i + ˆV ar p i + η ˆN pi = ˆV ar p i where a hat deotes a proportioal chage. Usig Propositio 3-iv i Alvarez ad Lippi 4 ad Lemma we have that these elasticities are related by: η = η ψ ad η σ = η ψ.thus η ψ ˆψ+ ηψ ˆN p i + ˆV ar p i +η ψ ˆN p i = ˆV ar p i. Rearragig ad cacelig terms: η ψ ˆψ + ηψ ˆN pi η ψ ˆV ar pi =. Dividig by η ψ we obtai that ˆψ = ˆV ar p i ˆN p i. Additioally, sice ȳ is a fuctio of ψ/b, the we ca write ψ/ = B V ar p i /N p i Ψ, l for some fuctio Ψ, l. That ψ as l follows because Lφ, as φ ad because, by Propositio 3-i i Alvarez ad Lippi 4, ȳ is icreasig i ψ ad has rage ad domai [,. For ψ = ad N p i > we obtai: = B V p N p i.this follows from usig the + square root approximatio of ȳ for small ψ + r, the expressio for N p i = σ /ȳ ad Propositio, i.e. N p i V ar p i = σ. To obtai the expressio for Ψ, we use Propositio 6 i Alvarez ad Lippi 4 where it is show that for = the Kur p i = 3/ +. 54

14 D The CPI respose to a moetary shock To compute the IRF of the aggregate price level we aalyze the cotributio to the aggregate price level by each firm. Firms start with price gaps distributed accordig to g, the ivariat distributio. The the moetary shock displaces them, by subtractig the moetary shock δ to each of them. After that we divide the firms i two groups. Those that adjust immediately ad those that adjust at some future time. Note that, for each firm i the cross sectio, it suffices to keep track oly of the cotributio to the aggregate price level of the first adjustmet after the shock because the future cotributios are all equal to zero i expected value. Let g p;, /σ, ȳ be the desity of firms with price gap vector p = p,..., p at time t =, just before the moetary shock, which correspods to the ivariat distributio with costat moey supply. The desity g equals the desity f of the steady state square orms of the price gaps give by Lemma 3 evaluated at y = p + + p times a correctio factor: 9 g p,..., p ;, σ, ȳ = f p + + p ;, σ, ȳ Γ / 37 π / p + + p / To defie the impulse respose we itroduce two extra pieces of otatio. First we let { p t, p,..., p t, p} the process for idepedet BM, each oe with variace per uit of time equal to σ, which at time t = start at p, so p i, p = p i. We also defie the stoppig time τp, also idexed by the iitial value of the price gaps p as the miimum of two stoppig times, τ ad τ p. The stoppig time τ deotes the first time sice t = that jump occurs for a Poisso process with arrival rate per uit of time. The stoppig time τ p deotes the first time that pt, p > ȳ. Thus τp is the first time a price chage occurs for a firm that starts with price gap p at time zero. The stopped process pτ, p is the vector of price gaps at the time of price chage for such a firm. The impulse respose for the aggregate price level ca be writte as: Pt, δ; σ,, ȳ = Θδ; σ,, ȳ + t θδ, s; σ,, ȳ ds, 38 where Θδ gives the impact effect, the cotributio of the moetary shock δ to the aggregate price level o impact, i.e. at the time of the moetary shock. The itegral of the θ s gives the remaiig effect of the moetary shock i the aggregate price level up to time t, i.e. θδ, sds is the cotributio to the icrease i the average price level i the iterval of times s, s + ds 9 See Sectio 5 of Alvarez ad Lippi 4 for this result ad the techical Appedix P i Alvarez, Le Biha, ad Lippi 6 for a derivatio. 55

15 from a moetary shock of size δ. Figure 3 displays several examples of impulse resposes the figures plots output, i.e. δ P/ɛ. The fuctios θ ad Θ are readily defied i terms of the desity g, the process { p} ad the stoppig times τ: Θδ; σ,, ȳ p ιδ ȳ δ j= p j g p;, σ, ȳ dp dp ad θδ, t; σ,, ȳ is the desity, i.e. the derivative with respect to t of the followig expressio: p ιδ <ȳ E [ j= p j τp, p {τp t} p = p ιδ g p;, σ, ȳ dp dp where ι is a vector of oes. This expressio takes each firm that has ot adjusted prices o impact, i.e. those with p satisfyig p ιδ < ȳ, weights them by the relevat desity g, displaces the iitial price gaps by the moetary shock, i.e. sets p = p ιδ, ad the looks a the egative of the average price gap at the time of the first price adjustmet, τp, provided that the price adjustmet has happeed before or at time t. We make 3 remarks about this expressio. First, price chages equal the egative of the price gaps because price gaps are defied as prices mius the ideal price. Secod, we defie θ as a desity because, strictly speakig, there is o effect o the price level due to price chages at exactly time t, sice i cotiuous time there is a zero mass of firms adjustig at ay give time. Third, we ca disregard the effect of ay subsequet adjustmet because each of them has a expected zero cotributio to the average price level. Fourth, the impulse respose is based o the steady-state decisio rules, i.e. adjustig oly whe y ȳ eve after a aggregate shock occurs. Give the results i Propositio 3 -Propositio 4 we ca parametrize our model either i terms of,, σ, ψ/b or istead parametrize it, for each, i terms of the implied observable statistics N p i, Std p i, l. These propositios show that this mappig is ideed oe-to-oe ad oto. We refer to l as a observable statistic, because we have show that the shape of the distributio of price chages depeds oly o it. Propositio Cosider a ecoomy whose firms produce products ad with steady state statistics N p i, Std p i, l. The cumulative proportioal respose of the aggregate price level t periods after a oce ad for all proportioal moetary shock of size δ ca be obtaied from the oe of a ecoomy with oe price chage per period ad with uitary stadard deviatio of price chages as follows: P t, δ ; N p i, Std p i = Std p i P 56 t N p i, δ Std p i ;,. 39

16 This propositio exteds the result of Propositio 8 i Alvarez ad Lippi 4 to the case of l /N p i >. 3 The proof proceeds by verificatio. It is made of three parts. First we itroduce a discrete-time, discrete-state versio of the model. Secod we show the scalig of time with respect to N a, ad fially the homogeeity of degree oe with respect to Std p i ad δ. The step by step passages of the proof are reported i the techical Appedix P i Alvarez, Le Biha, ad Lippi 6. The propositio establishes that the shape of the impulse respose is completely determied by parameters: ad l, whose comparative static is explored i Figure 3. Ecoomies sharig these parameters but differig i terms of N p i or Std p i are immediately aalyzed by rescalig the values of the horizotal ad/or vertical axis. I particular, a higher frequecy of price adjustmets will imply that the ecoomy travels faster alog the impulse respose fuctio this is the sese of the rescalig the horizotal axis. Istead, the effect of a larger dispersio of price chages is see by rescalig the moetary shock δ by Std p i ad by a proportioal scalig of the vertical axis. A further simplificatio to the last result is give by ext corollary, showig that for small values of the moetary shocks oe ca overlook the scalig by Std p i so that, for a give ad l determiig the shape, the most importat parameter is the frequecy of price chages N p i : Corollary For small moetary shocks δ >, the impulse respose is idepedet of Std p i. Differetiatig equatio 39 gives: P t, δ ; N p i, Std p i = δ δ P t N p i, ;, + oδ for all t > ad, sice fȳ =, the the iitial jump i prices ca be eglected, i.e.: P, δ ; N p i, Std p i Θ,l δ; Std p i = oδ. E A ecoomy with heterogeous sectors Assume that there are S sectors, each with a expediture weight es >, ad with differet parameters so that each has Ns price chages per uit of time, ad a distributio of price chages with kurtosis Kurs. I this case, after repeatig the argumets above for each sector ad aggregatig, we obtai that the area uder the IRF of aggregate output for a small 3 The proof i Alvarez ad Lippi is costructive i ature, exploitig results from applied math o the characterizatio of hittig times for browia motios i hyper-spheres, which is ot loger valid for >. Here we use a differet strategy which relies o limits of discrete-time, discrete state approximatios. 57

17 moetary shock δ is Mδ = δm = δ 6 ɛ s S es Ns Kurs = δ 6 ɛ D s S ds Kurs 4 where D is the expediture-weighted average duratio of prices D s S ad the ds are weights takig ito accout both relative expeditures ad duratios. I es NsD the case i which all sectors have the same duratios the ds = es ad M is proportioal to the kurtosis of the stadardized data. Likewise, the same result applies if all sectors have the same kurtosis. 3 I geeral, if sectors are heterogeous i the duratios or expeditures, the the kurtosis of the sectors with loger duratio or expeditures receive a higher weight i the computatio of M. es Ns For the Frech data, computatio of the duratio weighted kurtosis i equatio 4 icreases the estimated cumulative effect by about 5%, reflectig a correlatio betwee the kurtosis ad the duratio of price chages. F Frequecy of price chages i Retail vs. Wholesale I this appedix we documet that wholesale prices are as sticky as retail prices for a broad cross sectio of products sold i grocery stores. For wholesale price we use PromoData, a dataset o maufacturer prices for packaged foods from grocery wholesalers the largest wholesaler i each locatio. PromoData provides the price per case charged by the maufacturer to the wholesaler for a UPC i a particular day, for 48 markets, over the period 6-. The data icludes iformatio o almost 9 product categories ad more tha 5, UPC Market products, ad cotai iformatio o both base prices ad trade deals discouts offered to the grocery wholesalers to ecourage promotios. We compute the frequecy of price chages usig base prices excludig trade deals as well as icludig trade deals. 3 data. The frequecy of price adjustmet at the retail level is computed usig the IRI Symphoy The dataset cotais weekly scaer price ad quatity data coverig a pael of stores i 5 metropolita areas from Jauary to December, with multiple chais of retailers for each market. The dataset cotais aroud.4 billio trasactios from over 3 The effect of heterogeeity i N p i o aggregatio is well kow, so that D is differet from the average of N p i s, see for example Carvalho 6 ad Nakamura ad Steisso. 3 I PromoData firms report oly the dates i which their prices chage. We thus assume that the price is costat betwee reportig dates. We discard the last price ucompleted spell ad cosider products with at least two price chages. The frequecy of adjustmet is computed at the weekly level for comparability with the retail data sets eve though our data may have a higher frequecy. The frequecy of adjustmet is computed for each product i.e. UPC x Market give that the data is ot at the store level ad the aggregated usig equal product weights. 58

18 Table 3: Weekly Frequecy of Price Adjustmet - Wholesale vs Retail Level Data Period Frequecy excl. Sales Frequecy All Products PromoData Wholesale IRI Symphoy Retail -.. All Products 6 - PromoData Wholesale IRI Symphoy Retail Coffee PromoData Wholesale IRI Symphoy Retail -..9 RMS The table reports the weekly frequecy of price adjustmet usig three datasets: Nielse s PromoData, IRI Symphoy, ad Nielse s Retail Scaer RMS data. The frequecy of adjustmet is computed at the product level ad the aggregated across products usig equal weights. 7, UPCs ad aroud 3, stores. Goods are classified ito 3 geeral product categories ad a sales flag is provided whe a item is o discout thus we compute the frequecy both icludig ad excludig sales as i Sectio.. To correct for measuremet error due to compositio ad time aggregatio we oly retai price chages withi the iterval. p i log/3. Fially, to compare with ad exted Nakamura ad Steisso 8, we compute the frequecy of price chages for coffee usig data o retail prices ad sales from the Retail Scaer Data RMS by Nielse. Our data is at the week-productstore level for the period of 6-3. The structure of the dataset is the same as the IRI Symphoy data except that the RMS does ot provide a sales flag, ad covers about cities. Table 3 summarizes the mai fidigs of this measuremet exercise. The weekly frequecy of price adjustmet sales excluded for the etire wholesale data PromoData is.9 per week which compares with a mea frequecy of adjustmet of about. per week i the retail IRI data. Frequecies of comparable magitude are detected across samples from differet segmets of the distributio chai, as well as for differet items coffee ad beer, ot reported i the samples that exclude sales. Icludig sales makes the frequecy of adjustmet i retail somewhat higher tha the frequecy i wholesale. 59

19 G Simple special cases of Propositio 6 This sectio discusses some limitig cases i which tractable closed form expressios for the cumulative effect M as well as the frequecy ad kurtosis of price adjustmets ca be derived. The first two cases we illustrate assume either = or : we derive the implicatios for the cumulative output effect while cosiderig the full rage of values for l, ad keepig the frequecy of price chages costat. The last case restricts attetio to l = or l = but allows for ay value of. G. Aalytical computatio of M i the case of = We give a aalytical summary expressio for the effect of moetary shocks i two iterestig cases, those for oe product, i.e. =, ad those for the large umber of product, i.e. =. The summary expressio is the area uder the impulse respose for output, i.e. the sum of the output above steady state after a moetary shock of size δ >, which we deote as: M δ = /ɛ [δ P δ, t dt 4 where /ɛ is related to the ucompesated labor supply elasticity ad P δ, t is the cumulative effect of moetary shock δ i the log of the price level after t periods. For large eough shocks, give the fixed cost of chagig prices, the model display more price flexibility. Because of their promiece i the literature, ad because of realism, we cosider the case of small shocks δ by takig the first order approximatio to equatio 4, so we cosider M δ M δ. For the case of = we obtai a aalytical expressio which, after ormalizig by N p i depeds oly o /N p i. Thus as /N p i rages from to the model rages from a versio of the meu cost model of Golosov ad Lucas to a versio usig Calvo pricig. The aalytical expressio is based upo the followig characterizatio: p δ M δ = /ɛ mp gp + δ dp 4 p where p is the price gap after the moetary shocks ad where mp gives the cotributio to the area uder the IRF of firms that start with price gap, after the shock, equal to p. Sice the moetary shock happes whe the ecoomy is i steady state, the distributio right after the shock has the steady state desity h displaced by δ. Immediately after the shock the firms with the highest price gap have price gap p δ. Note that the itegral i equatio 4 does ot iclude the firms that adjust o impact, those that before the shock 6

20 have price gaps i the iterval [ p, p δ, whose adjustmet does ot cotribute to the IRF. The defiitio of m is: [ τ mp = E pt dt p = p where τ is the stoppig time deotig the first time that the firm adjusts its price. This fuctio gives the itegral of the egative of the price gap util the first price adjustmet. This expressio is based o the fact that those firms with egative price gaps, i.e. markups, cotribute positively to output beig i excess of its steady state value, ad those with high markups cotribute egatively. Give a decisio rule summarized by p we ca characterize m as the solutio to the followig ODE ad boudary coditios: mp = p + σ m p for all p [ p, p ad mp = otherwise. low The solutio for the fuctio m is: mp = p + p e φ p p e φ p p e φ e φ for all p [ p, p. φ p /σ. We the have: Mδ M δ = δ/ɛ p p mp g p dp = δ/ɛ p mp g p dp sice m pg p =. The last equality uses that m is egative symmetric, i.e. mp = m p, ad that g is symmetric aroud zero. Usig the expressio for g i Sectio 3. g φ p = p e φ e φ p p + e φ p p for p [, p. we obtai: δm = δ ɛ e φ e φ + φ e φ + e φ. Usig the expressio for N p i for the = ad simple algebra we ca rewrite it as: δm = δ ɛ e φ + e φ N p i e φ + e φ e φ + e φ φ 6

21 which yields the cumulative output effect of a small moetary shock of size δ. 33 Kurtosis. We ow verify that the expressio ca be equivaletly obtaied by computig the kurtosis, as stated i Propositio 6. For otatio coveiece let x φ. Usig the distributio of price chages derived i Sectio 3. ad the defiitio of kurtosis we get Kur p i = +x l + l x 4 x e x/ e x/ l = + x e x e + l x x +x 4 + x 4 l e x/ e x/ l l x + + x e x e l x l Recall from Sectio 3. that l = ex +e x e x +e x so that, after some algebra Kur p i = 6 e x + e x e x + e x e x + e x x It is immediate that the kurtosis ad the cumulative output effect satisfy Propositio 6. G. Aalytical computatio of M i the case of = Defie Y t, δ i= [p i t δ = Y t, δ p it + δ. i= where the p i t are idepedet of each other, start at p i = ad have ormal distributio with E [p i t = ad V ar [p i t = σ t. The, by a applicatio of the law of large umbers, we have: Lettig Ȳ Y t, δ = Y t, + δ = tσ + δ lim ȳ/ we ca represet the steady state optimal decisio rule as adjustig prices whe t, the time elapsed sice last adjustmet, attais T = Ȳ /σ. compute the desity of the distributio of products idexed by the time elapsed sice the last adjustmet t ad, abusig otatio, we deote it by f. This distributio is a trucated 33 As a check of this formula compute the case for φ =, i.e. the cumulative output for the Golosov-Lucas model. I this case we let = ad p >. I this case we have: mp = p p 3σ + p3 3σ. Also g p = / p for p, p, so we have: M δ = δ ɛ 3σ p p [ p p + p 3 δ dp = [ p4 ɛ 3σ p + p4 = 4 δ p ɛ 3σ 8 = δ ɛ which is the same value obtaied by takig the limit for φ i the geeral expressio above. N p i 6 We 6

22 expoetial with decay rate ad with trucatio T, thus the desity is: e t ft = for all t [, T. e T The expected umber of price chages per uit of time is give by the sum of the free adjustmets ad the oes that reach T, so N p i = + ft = [ + e T = e T e T Note that, usig the defiitio of T give above, T = Ȳ /σ the parameter which idexes the shape of f ad of the distributio of price chages. Sice this figures promietly i this expressios we defie: φ T = Ȳ σ. which is cosistet with the defiitio of φ i Propositio 3. Usig this defiitio we get: l = N p i = e φ ad thus N p i = Impulse Respose of Prices to a moetary Shock. e φ We ca ow defie the impulse respose. Note that after the moetary shock firms that have adjusted their prices t periods ago, i average will adjust their price up by δ. This highlights that as there is o selectio. Now we tur to the characterizatio of the impact effect Θ. I this case we have Y t, δ = Y t, + δ = tσ + δ Ȳ = σ T t T δ /σ. Thus the impact effect is: T Θδ = δ ftdt = δ e T + σ δ e T T δ /σ Usig that N p i V ar p i = σ we ca write: Θδ = δ + δ e κ+ N p i δ V ar p i e κ = δ + δ σ δ e κ = δ e κ+ e κ e κ N p i e N p i /N p i δ V ar p i 63

23 Note that δ δ Std p lim Θδ = i as /N p i as /N p i ad i geeral Θδ e /N p i = δ wheever δ < Std p i. δ N p i V ar p i N p i + < /N p i δ V ar p i N p i θt = δe t [f T δ /σ t + T δ /σ t fsds = δ e t e T. We ca iterpret θtdt as θt times the umber of firms that adjust its price at times t, dt. This is the sum of two terms. The first term is the fractio that adjust because they hit the boudary betwee t ad t + dt. The secod term is the fractio that have ot yet adjusted times the fractio that adjust, dt due to a free opportuity. Both terms are multiplied by e t to take ito accout those firms that have received a free adjustmet opportuity before after the moetary shock but before t. Thus we have: t e s P t, δ = Θδ + δ e N p i t N p i e ds = Θδ + δ T /N p i Usig P we ca compute the IRF for output, ad a summary measure for it, amely the area below it: M δ = ɛ T [δ P δ, t dt δ ɛ N p i [ + φ e φ e φ where the approximatio uses the expressio for small δ, i.e. its first order Taylor s expasio. Kurtosis. For completeess we also iclude here a expressio for the kurtosis of the distributio of price chages i the case of =. Price chages are distributed as: E [ p i = σ /N p i = σ N p i = T σ T N p i = T σ T N p i 64

24 E [ p i 4 = 3 N p i T σ t e t dt + e T = 3σ 4 T [ e T T T + + T + Some algebra shows that kurtosis is the give by: E [ p i 4 E [ p i = 6 e φ + φ e φ N p i It is immediate to use the expressios above to verify Propositio 6. 3 σ T N p i G.3 Aalytical computatio for l = or l = ay. For l =, or equivaletly =, we use the result i Alvarez ad Lippi 4 for T + y = ȳ y + σ gives: M = ɛ ȳ [ ȳ y y + σ fy dy ad usig the followig expressio for f from Alvarez ad Lippi 4 : fy = [logȳ logy if =, ad ȳ fy = ȳ [ ȳ y otherwise gives that: which verifies the equality i Propositio 6. M = ȳ ɛ 4σ = Kurt p i ɛ 6 N p i For < < ad l =, with > ad σ >, usig Propositio 3 it must be the case that ȳ =. I this case, N p i = l =, ad the distributio of price chages is idepedet across each of the products, ad give by a Laplace distributio, which has kurtosis 6. Likewise T + y = / for all y. Thus, usig equatio 6 we obtai the desired result. 65