Small and large price changes and the propagation of monetary shocks

Transcription

1 Small and large price changes and the propagation of monetary shocks Fernando Alvarez, Hervé Le Bihan, Francesco Lippi University of Chicago & NBER, Banque de France, EIEF & University of Sassari Banque de France June 2014 ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 0 / 16

2 Intro Cumulative output after monetary shock: M output deviation from steady state (in %) δ ɛ M(δ) 1 ɛ 0 (δ P t ) dt time t ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 1 / 16

3 Intro Range of effects: M Calvo / M GL 6 Percent deviations from the initial steady state 0.7 FIGURE 5: OUTPUT RESPONSES IN MENU COST AND CALVO MODELS Benchmark Calvo Benchmark MC Calvo with fixed factor MC with fixed factor source: Golosov-Lucas Quarters More Models: Reis / Taylor 1980, Nakamura-Steinsson 2010, Midrigan 2011 All models same N ( p i ) and Std ( p i ) ; differences due to selection effect ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 2 / 16

4 Our contribution Intro Large class of models (i.e continuous shocks): (random) menu cost, multi-product, rational inattentiveness. M(δ) cumulative IRF of output to a (small) monetary shock δ M(δ) δ Kur( p i) 6 N ( p i ) Frequency of price changes N ( p i ) has a first order effect (cross-section) Kurtosis of price changes Kur( p i ) has a first order effect Result in the spirit of Sufficient Statistic Approach" Gather and evaluate empirical measures of Kur( p i ) 4.5 ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 3 / 16

5 The evidence Micro evidence on Price Changes: Summary We explore several datasets CPI (French, US, Norway), scanner price (US), Internet scraped data Excess Kurtosis due to small and large p i (Kashyap, 1995) Excess Kurtosis attenuated but still positive after correcting for: (1) heterogeneity standardize price changes at the good store level (2) measurement error compare datasets and trimming Bottomline: distribution of p i peaked in both the US and France between Normal (kur 4 in US) and Laplace (kur 5 in FR) ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 4 / 16

6 The evidence French CPI data size distribution; Standardized and trimmed; 1.5 million obs. 1 Data Standardized Normal Standardized Laplace N ( p i ) = 2 per year ; std( p i ) = 16% ; kurtosis = 8.9 ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 5 / 16

7 The evidence Safeway data: Eichenbaum et al (2011) scanner data Size distribution of non-zero p i (based on average weekly prices) All p i trim p i < 1 dollar cent Density dp Density dp kurtosis = 4.9 ; 11 million obs. kurtosis = 3.8 ; 8.5 million obs. Based upon: Average Weekly data. Lots of price changes smaller than 1 cent! (left panel) ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 6 / 16

8 The firm problem Model has 4 primitive parameters ψ B, λ, σ2 and n production cost each product: random walk w/volatility σ 2 dt. p i (t) (log) percentage deviation from static optimal markup over cost. 2nd order approx. to period profit gives: n i=1 B [p i (t)] 2, B = (1/2) η (η 1) where η elasticity of demand. firm maximizes expected net discounted (at r) profits. multi-product: simultaneous adjustment of n products sold by firm. random menu cost: adjust n prices paying = { ψ with probability 1 λ dt or 0 with probability λ dt. sequence prob. ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 7 / 16

9 The firm problem Simplest case when n = 1 and λ 0 Bellman equation r V (p) = B p 2 + λ [V (0) V (p)] + σ2 2 V (p), for p ( p, p), value matching and smooth pasting conditions are: V ( p) = V (0) + ψ, V ( p) = 0, p ( 6 ψ σ 2 ψ B B ) 1 4 (r + λ) if if ψ B σ2 (r + λ) 2 is small ψ B σ2 (r + λ) 2 is large Threshold rule p yields cross-sectional model predictions ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 8 / 16

10 Cross-section predictions Cross section predictions for n 1 All prices change if p(τ) 2 hits ȳ or when free-chance arrives. At such times price changes are p i (τ) = p i (τ), i.e. reset gaps to zero. Use properties of p(τ) 2 [0, ȳ) and statistical tools to characterize: scale and frequency of p i : Std( p i ), N ( p i ) = l λ N( p i ) size-distribution of p i : w( p i ): shape depends ONLY on n and l one-to-one mapping (n, λ, ψ/b, σ 2 ) and (n, l, Std( p i ), N ( p i ) ). ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 9 / 16

11 Cross-section predictions for n = 1, l = λ/n = 0.05, N = 2, Std( p i ) = Density l = kur = Size of price changes p i Size distribution w( p i ) for n = 1 model ; shape depends ONLY on l ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 10 / 16

12 Cross-section predictions for n = 1, l = λ/n = 0.80, N = 2, Std( p i ) = Density 4 l = kur = Size of price changes p i Size distribution w( p i ) for n = 1 model ; shape depends ONLY on l ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 11 / 16

13 Cross-section predictions for n = 1, l = λ/n = 0.99, N = 2, Std( p i ) = Density 4 3 l = kur = Size of price changes p i Size distribution w( p i ) for n = 1 model ; shape depends ONLY on l ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 12 / 16

14 Cross-section predictions Kurtosis Kur( p i ) function only of (n, l) 6 =1 =2 5.5 =10 n = 5 Calvo Kurtosis Taylor Midrigan Calvo Plus (NS) 1 Golosov Lucas Fraction of free adjustments: l Kur( p i ) [1, 6], an increasing function of (n, l) ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 13 / 16

15 Impulse response GE model and real effects of money shocks GE structure is version of Golosov-Lucas, Money in UF, no inflation labor only, CRTS, (uncompensated) labor supply elasticity 1/ɛ. P(δ, t): aggregate prices t periods after once a for all money shock δ. include only GE feedback effects that are first order on (δ, ψ) M area under output s IRF to a monetary shock δ: M (1/ɛ) 0 [δ P(δ, t)] dt ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 14 / 16

16 Impulse response Key idea to simplify the problem Expected cumulated output deviation from st st of firm w/ price gap p 0 : m(p 0 ) = 1 ( τ ) ɛ E ( p(t)) dt p 0 τ: stopping time associated with optimal decision rule. Useful because m(p 0 ) is characterized by an ODE example for n = 1: λm(p) = p + m (p) σ2 2 Using invariant distribution of price gaps g(p): M(δ) = Main result: for small shocks M(δ) δ 1 ɛ p p 0 m(p δ) g(p) dp Kurt( p i ) 6 N a for all n 1 and l [0, 1] ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 15 / 16

17 Impulse response Cumulated output effect relative to Calvo details Output effect relative to Calvo Taylor n =1 n =2 n =10 n = Midrigan Calvo Plus (NS) Calvo 0.2 Golosov Lucas Fraction of free adjustments: l Kurtosis / 6 One curve for each n, no other parameters; infinite slope at l = 1 Differences are due to selection on time" and size" of adjustments. details ψ fixed cost, B curvature, n # products λ pr. free adjustment, σ cost volatility l fraction free-adjustments 16 / 16

18 Remaining topics Additional analysis Lack of sensitivity to Inflation Aggregation across sectors. Free vs Cheap random cost. Rational Inattentiveness: no menu cost. Fat-tailed shocks. Large Monetary Shocks. Implied menu costs

19 Inflation sensitivity Lack of sensitivity to Inflation µ Static target" prices have drift µ, all price gaps drift down p i (t) = µ t + σw i (t) + j:τ j <t p i(τ j ) all t 0, i = 1,..., n. Optimal decision rule are different (no closed form) State is entire vector p, not just y = p 2. Prices are not reset to static target at adjustment. Inaction set I is not a hyper-sphere. GE model: nominal rate r + µ, wages grow at µ.

20 Inflation sensitivity Lack of sensitivity to Inflation µ Inflation has only second order effect around µ = 0 on entire hazard rate h and frequency of price changes N ( p i ), marginal distribution of absolute value of price changes w( p i ), all centered even moments of marginal price changes (e.g. Kur( p i )). area under output IRF small monetary shock: M (0) results on M (0) due to symmetry of µ around zero & of (µ, δ) around (0,0). Thus expression for holds for small inflation rates: M(δ; µ) δ ɛ Kur( p i ; µ) N ( p i ; µ) 6

21 Heterogenous sectors Aggregation across heterogeneous sectors Let e(s) the expenditure share of sector s with different parameters. Allow different, N(s), Std(s) and Kur(s) by sector s For small δ shocks aggregation across sectors yields: M(δ) 1 δ 6 ɛ s S e(s) N(s) Kur(s) If Kur(s) the same across sectors: the model aggregates using expenditure weighted by duration. If Kur(s) varies across sectors: needs to consider its covariation. In French data we found about 15% higher effect due to this effect.

22 Cheap cost Cheap vs free adjustment cost Take the case of n = 1 product. cheap random menu cost: adjust the price paying = { ψ b ψ with probability 1 λ dt or with probability λ dt. Cheap adjustment fraction b [0, 1) of normal cost ψ. Optimal policy: two thresholds 0 < p < p, adjust price if square price gap = [ p(t) ] 2 { p p and cost ψ and cost b ψ Eliminates price changes smaller than p, & hence decreases Kur( p i ). Result M δ ɛ Kur( p i ) 6 N( p i ) still holds.

23 Cheap cost Fraction of cheap price changes for several p and p Fraction of cheap price changes, keeping σ and λ Fraction of cheap price changes p =0.025 p =0.05 p =0.1 p =0.15 p = p as fraction of p

24 Cheap cost Kurtosis for several p and p Kurtosis ( p)keeping σ and λ fixed p =0.025 p =0.05 p =0.1 p =0.15 p =0.2 kurt ( p) p as fraction of p

25 Oservation cost model Rational Inattentiveness (no menu cost): Caballero, Reis, Carvalho-Schwartzman, Alvarez-Lippi-Paciello. Firms observe price gap only if they pay a random observation costs. Optimal decision rules: details set times τ until next review = and price adjustment decision con t (cannot) depend on (unobserved) current price gaps. Randomness of expected observation cost implies random times τ. Models with n = have similar formal properties.

26 Oservation cost model Rational Inattentiveness (no menu cost): Caballero, Reis, Carvalho-Schwartzman, Alvarez-Lippi-Paciello. Firms observe price gap only if they pay a random observation costs. Optimal decision rules: details set times τ until next review = and price adjustment decision con t (cannot) depend on (unobserved) current price gaps. Randomness of expected observation cost implies random times τ. Models with n = have similar formal properties. Monetary shock δ learned only at times when firms observe their cost. Allows for any value of Kur( p i ), even larger than 6. Result M δ ɛ Kur( p i ) 6 N( p i ) still holds.

27 Fat tailed shocks Model with fat tailed shocks Process for cost: BM W + Poisson counter N w/ intensity λ dp i (t) = σ dw i (t) + ξ i (t) dn(t) for i = 1,..., n distribution of fat-tailed shock: 0 < ξ inf ξ with Γ probability = 1 Results: 1 If the ξ large enough, then threshold ȳ is the same as in baseline model. 2 Fat tails contributes to kurtosis by mostly adding large price changes. 3 "Lack of identification". (almost) anything goes if Γ unrestricted and ψ small.

28 large shocks Effect depends on size of the monetary shock δ δ smallest once-and-for-all monetary shock that gives full price flexibility. Depends on Std( p i ) and l: ȳ L δ = 2 n = 2 Std( p 1 (l; n) i), l where L( ; n). Fixing any n 1, the ratio δ is a strictly increasing function of l 2 Std ( p i )) ranges from 1 to as l goes from 0 to 1.

29 large shocks Effect depends on size of the monetary shock δ Minimum δ for price flexibility increases with l > 0. Figure: Case of l = 0, so that range is 2 Std[ p] PRICE SETTING WITH MENU COST 123 FIGURE 6. Cumulative output effect M(δ).ParametersareNa = 1 and Std( pi) = 0 10.

30 Implied menu costs Implied menu costs (for r 0) Fixing n and l (0, 1), the menu cost ψ 0 : Menu cost per product = ψ n = B Var( p i) N ( p i ) Ψ (n, l) is only a function of (n, l). Total cost: B = η(η 1)/2 or markup m 1/(η 1) then Ψ (n, l) Yearly costs of price adjustment Yearly revenues = 1 2 Var( p i ) m (1 l) Ψ (n, l) examples

31 Implied menu costs The cost of price adjustment 25 1 Menu cost / profits * n =1 n =2 n =10 n = Yearly costs of adjustment / revenues * n =1 n =2 n =10 n = Share of free adjustments: l Share of free adjustments: l Levy et al (QJE, 1997) avg. cost per year around 4-5% of profits. Levy et al (QJE, 1997) avg. cost per year around 0.7% of revenues.

32 APPENDICES

33 more-micro-evidence French CPI data for p i CPI Data Benchmarks all records exc.sales Normal Laplace Frequency of price changes Moments of standardized price changes: z Kurtosis Moments for the absolute value of standardized price changes: z Average: E ( z ) Fraction of observations < 0.25 E ( z ) Fraction of observations > 2 E ( z ) Number of obs. with p 0 1,544,829 1,080,183 cross-section heterogeneity standardized price changes z i,j,t = p i,j,t m ij σ ij i-good category (270), j-outlet type (11), t-time (120 months) bottom-line: distribution of the standardized p closer to Laplace than Normal back

34 more-micro-evidence Figure: Histogram of Standardized Price Adjustments: French CPI All data 1 Data Standardized Normal Standardized Laplace Excluding sales 1 Data Standardized Normal Standardized Laplace

35 more-micro-evidence Measurement error: CPI vs Scraped Data (France) Table: Matching CPI vs. the BPP by store x products Statistic BPP BPP CPI BPP CPI retailer 1 retailer 5 Hypermarkets retailer 4 Large ret. electr. duration Statistics for standardized price changes: z mean z % below 0.5 mean z % below 0.25 mean z kurtosis of z back

36 more-micro-evidence Measurement error: CPI vs Scanner data France US (scanner data) Cavallo-Rigobon EJR Midrigan CPI BPP Safe-way IRI Nielsen Statistics for standardized price changes: z mean of z % below 0.50 mean z % below 0.25 mean z kurtosis of z: Measurement error model: measured changes = true (u) + error (ɛ) true u w/ pr θ and std σ u, error ɛ w/ pr 1 θ and std σ e : lim σe 0 kurt = kurtu θ bottom-line: comparison of BPP vs CPI suggests θ = 1/2 and kurt (3, 5) back

37 more-micro-evidence Pooled Standardized p: US vs France CPI 0.25 US CPI France CPI Laplace Normal Fraction Size of price changes

38 more-micro-evidence Kurtosis on U.S. CPI Klenow-Krystov: standardized and discretized = (previous graph) French CPI: standardized and discretized = (previous graph) Klenow-Malin: raw data = 10 (posted prices), 17.4 (regular prices) Vavra: 6.4 (trimmed) and 4.9 (also standardized) back

39 setup Firm s problem: approx. to CES demand + CRTS V (p) = where min {τ j, p(τ j )} j=1 E 0 ( n ) e rt B pi 2 (t) dt + e rτ j ψ1 dn (t)=0 p(0) = p i=1 p i (t) = p i (0) + σw i (t) + p i (τ j ) for all t 0 and i = 1, 2,..., n, j:τ j <t j=1 dp i = σ dw i : n Independent Brownian Motions (prod. shocks). stopping times τ j and adjustments p i (τ j ) all i = 1,.., n and j = 1, 2,... Poisson dn (t) = 1 with probability λdt, and zero otherwise pay fixed cost ψ1 dn (t)=0 and adjust prices of all products p. back to setup derivation from first principles simplifying assumptions alternative interpretation

40 Analytics ell back Fraction l λ/n ( p i ) as function of φ, n : [ φ 1 + ( i i=1 k=1 L(φ, n) [ 1 + φ 1 + ( i i=1 k=1 n (k+1) (n+2k) n (k+1) (n+2k) Density f ( ) of invariant distribution of y = p 2 f (y) = ) φ i ] ) φ i ] where φ λ ȳ n σ 2 ( ) λy ( n 4 1 2) [ ( ) ( )] λy λy 2σ 2 C 1 I ν 2 2σ 2 + C 2 K ν 2 2σ 2 I ν ( ) and K ν ( ) modified Bessel functions of the first & second kind, C 1, C 2 arbitrary constants (so that integrates to one and ȳ is an exit point), and ν = n 2 1. Note that argument of f ( ) is λy 2σ 2, which has the same units as φ

41 Bellman and boundaries Value fct. v(y) and optimal threshold ȳ Find value function v( ) : R + R + and value ȳ > 0. Bellman ODE in inaction region [0, ȳ): r v(y) = B y + n σ 2 v (y) + 2 σ 2 y v (y) + λ(v(y) v(0)), for y (0, ȳ). At the boundary ȳ: v(ȳ) = v(0) + ψ and v (ȳ) = 0 Optimal return at y = 0 v (0) 0. Optimal decision rule: ȳ same with (r, λ) and (r + λ, 0). 2(n + 2) σ 2 ψ/n B if ψ B σ2 (r + λ) 2 is small (option value), ȳ/n ψ/n B (r + λ) if ψ B σ2 (r + λ) 2 is large (no option value). back to overview

42 Characterization of M Area under Output s IRF of a monetary shock δ Area under IRF of output M(δ) = 1 ɛ 0 [ δ P(δ, t) ] dt Expected cumulated st. st. output deviation of firm w/ price gap p: [ m (p 1,..., p n ) = 1 τ n n E p i (t) dt ] p(0) = p where is the τ stopping time associated with optimal decision rule. Using invariant distribution of price gaps g(p): M(δ) = 1... m (p 1 δ,..., p n δ) g (p 1,..., p n ) dp 1... dp n nɛ For a small shock: M n,l (δ) δ M n,l (0) with: M (0) = 1 n m... (p 1,..., p n ) g (p 1,..., p n ) dp 1... dp n nɛ p i back to n=1 i=1 0 i=1

43 Characterization of M small monetary shocks δ Using the scaling properties of P(δ, t) we have (for fixed n, l): M ( δ ; N ( p i ), Std[ p i ] ) = Std[ p ( ) i] N ( p i ) M δ Std[ p i ] ; 1, 1 Taylor expansion of M(δ) around δ = 0: M (δ; N ( p i ), Std[ p i ]) δ M (0; 1, 1) N ( p i ) which is independent of Std[ p i ]. CPI

44 Characterization of M small monetary shocks δ Using the scaling properties of P(δ, t) we have (for fixed n, l): M ( δ ; N ( p i ), Std[ p i ] ) = Std[ p ( ) i] N ( p i ) M δ Std[ p i ] ; 1, 1 Taylor expansion of M(δ) around δ = 0: M (δ; N ( p i ), Std[ p i ]) δ M (0; 1, 1) N ( p i ) which is independent of Std[ p i ]. Main result: Notice: M (δ; N ( p i ), Std[ p i ]) δ M (0; 1, 1) N ( p i ) = δ ɛ Std( p i ) doesn t feature in expression for small shocks, Kur ( p i ) depends ONLY on n and l M depends on n and l ONLY though Kurtosis back to n=1 Kur ( p i ) 6 N ( p i ) CPI

45 Inflation sensitivity Sensitivity to Inflation µ Proposition Static target" prices have drift µ, all price gaps drift down p i (t) = µ t + σw i (t) + j:τ j <t p i(τ j ) all t 0, i = 1,..., n. Optimal decision rule are different (no closed form) State is entire vector p, not just y = p 2. Prices are not reset to static target at adjustment. Inaction set I is not a hyper-sphere. Inflation has only second order effect around µ = 0 on entire hazard rate h and frequency of price changes N ( p i ), marginal distribution of absolute value of price changes w( p i ), all centered even moments of marginal price changes (e.g. kurtosis). expected value function E [V (p)].

46 Inflation sensitivity Sensitivity to Inflation µ, Proposition Back Index statistics with inflation rate µ. (a) µ N ( p i) (µ) µ=0 = 0, and µ h (t, µ) µ=0 = 0 for all t 0, (b) (c) µ E [ p 1 i, µ] µ=0 = N ( p i ) (0) > 0 and [ ] µ E ( p i E[ p i ]) 2k, µ = 0, for k = 1, 2,..., µ=0 2 µ 2 E [ p i, µ] µ=0 = 0, (d) µ M (0; µ) µ=0 = 0.

47 Inflation sensitivity Table: Fraction of small price changes: US and French CPI Moments for the absolute value of price changes: p France US Normal Laplace Average p Fraction of p below 1% Fraction of p below 2.5% Fraction of p below 5% Fraction of p below (1/14) E ( p ) Fraction of p below (2.5/14) E ( p ) Fraction of p below (5/14) E ( p ) Number of obs 1,542,586 1,047,547 Data is NOT standardized

48 General Equilibrium Set Up General Equilibrium Set Up ( c(t) Lifetime Utility : e r t 1 ɛ 1 α l(t) + log M(t) ) dt 0 1 ɛ P(t) ( ) η 1 n CES aggregate : c(t) = ( Z ki (t) c ki (t) ) 1 1 η 1 η dk 0 i=1 Elasticities: Intertemporal 1 ɛ, Intratemporal η (= for firms k & products i). Linear technology c ki (t) = Z ki (t) l ki (t) and Z ki (t) = exp (σ W ki (t)). Adjustment Cost: if ψ l units of labor paid, then n prices can be changed. Equilibrium: constant nominal interest rate & wages W (t) = a M(t). back to firm-problem back IRF

49 Approximation and Interpretation of loss function Approximation and Interpretation of loss function Discounted nominal profit product i as function of (log) price gap p i : Π (p i, c(t)) W (t) e r t c(t) 1 ηɛ e η p i 0 = Π (0, c(t)) p i = 2 Π (0, c(t)) p i c(t) [ ] e p η i η nd order expansion around P, relative to frictionless profits. P i (t) = ˆP i (t) P i (t) P i (t) or in logs: p i (t) = % deviation from optimal price. 1 st order term vanishes, second order: B p(t) 2 with B = η(η 1) 2 Fixed cost frictionless profits n so ψ = Physical Cost in Numerarie back to firm-problem back IRF back to tricks to simplify

50 Approximation and Interpretation of loss function Useful simplifying assumptions for our problem Simplification on firm problem unit root shocks (no mean reversion): state summarized by gaps Simplification on eq. structure Linear leisure + log(m/p) + one-time shock wages proport. to money back to firm-problem

51 Comparison with rational inattentiveness model Comparison with rational inattentiveness model (time-dependent rule) As n our model predicts Similarities with an observation cost model (Caballero/Reis) Normal distribution of price changes time-dependent rule (spike in hazard at t = 1/N ( p i )) Uniform invariant distribution of y gaps Difference Response to shocks depends on size of shocks

52 Comparison with rational inattentiveness model Evidence on synchronization of price changes of similar goods within firms Lach & Tsiddon (96): within store different goods > across stores on same good. Dutta, Bergen, Levy & Veneable (99): day of week by type of good. Fisher & Konieczny (00): newspapers owns by same firm vs others. Chakrabarti & Sholnick (07): Amazon.com and BarnesNoble.com Midrigan (09): narrow UPC + logit using other product change. Battarai & Schoenle (10): logit using other product change. Cavallo (10): narrow UPC for online stores on 4 developing countries Size & frequency of p vs. # goods sold by firms (Battarai and Schoenle) back to model

53 Comparison with rational inattentiveness model back Fraction l λ/n ( p i ) as function of φ, n : [ φ 1 + ( i i=1 k=1 L(φ, n) [ 1 + φ 1 + ( i i=1 k=1 n (k+1) (n+2k) n (k+1) (n+2k) Density f ( ) of invariant distribution of y = p 2 f (y) = ) φ i ] ) φ i ] where φ λ ȳ n σ 2 ( ) λy ( n 4 1 2) [ ( ) ( )] λy λy 2σ 2 C 1 I ν 2 2σ 2 + C 2 K ν 2 2σ 2 I ν ( ) and K ν ( ) modified Bessel functions of the first & second kind, C 1, C 2 arbitrary constants (so that integrates to one and ȳ is an exit point), and ν = n 2 1. Note that argument of f ( ) is λy 2σ 2, which has the same units as φ

54 Comparison with rational inattentiveness model Define for n 2, distr. of price changes conditional on reaching y: ( ( ) ) 2 (n 3)/2 1 ω( p i ; y) = Beta( n 1 2, 2) 1 1 pi y if ( p y i ) 2 y 0 if ( p i ) 2 > y Distribution of price changes: mixture [ ] ȳ w( p i ) = ω( p i ; ȳ) (1 l) + ω( p i ; y)f (y)dy l for n 2. 0 Shape depends on (n, l). Let w( p i ; n, l, 1) density price changes p i : n goods, share l of free adjustments, unit standard deviation of price changes Std( p i ) = 1. This density is hod -1 in p i and S = Std( p i ): w (S p i ; n, l, S) = 1 S w ( p i; n, l, 1) for all S > 0. back

55 Comparison with rational inattentiveness model Kurtosis for selected values of l and n back (i) Kur( p i ) depends on two parameters: n and l λ N( p i ) (ii) Let ψ/b so that ȳ. Then N ( p i ) λ and Kur( p i ) 6 (Laplace) % of free adjustments: number of products n l λ/n ( p i ) % % % % % % %

56 Comparison with rational inattentiveness model Implied menu costs (examples closed form) Recall menu cost of one adjustments and total cost: Menu cost per product ψ n = B Var( p i) N ( p i ) Yearly costs of price adjustment Yearly revenues = 1 2 Var( p i ) m Ψ (n, l) (1 l) Ψ (n, l)

57 Comparison with rational inattentiveness model Implied menu costs (examples closed form) Recall menu cost of one adjustments and total cost: Menu cost per product ψ n = B Var( p i) N ( p i ) Yearly costs of price adjustment Yearly revenues = 1 2 Examples for extreme cases n = 1 and n = : Ψ (1, l) = 1 ( 1 [arcosh l 2 ) l 2 ( ) 1 l arcosh 1 l Var( p i ) m Ψ (n, l) (1 l) Ψ (n, l) ( ( ))] 1 coth arcosh, 1 l log (1 l) + l Ψ (n, l) l 2, as n back

58 Comparison with rational inattentiveness model Two product case (n = 2): shift each price gap by δ 1 price gap for good 2: p price gap for good 1: p 1 - Case of n = 1 all firms with low price adjust, and they all increase prices - Case of n = 2 only those with n i=1 (p i δ) 2 ȳ adjust, and only some increase both prices (S-W quadrant) back

59 Comparison with rational inattentiveness model IRF Prices: Impact effect and limiting cases Impact effect Θ(δ) is 2 nd order in δ. Flexible prices for large δ and l < 1.

60 Comparison with rational inattentiveness model IRF Prices: Impact effect and limiting cases Impact effect Θ(δ) is 2 nd order in δ. Flexible prices for large δ and l < 1. If l = 1 and any n: adjust only when free (Calvo) First that change prices t periods after shock change by δ in average: For small shock δ: P(δ, t) = δ (1 exp ( N ( p i ) t))

61 Comparison with rational inattentiveness model IRF Prices: Impact effect and limiting cases Impact effect Θ(δ) is 2 nd order in δ. Flexible prices for large δ and l < 1. If l = 1 and any n: adjust only when free (Calvo) First that change prices t periods after shock change by δ in average: For small shock δ: P(δ, t) = δ (1 exp ( N ( p i ) t)) If l = 0 and n = : adjust every 1/N ( p i ) periods, time dependent (Taylor) First that change prices t periods after shock change by δ in average: For small shock δ: P(δ, t) = δ max {N ( p i ) t, 1} back

62 Comparison with rational inattentiveness model Stopping time τ(p) & price gap p(t, p) for initial gap p P(t, δ) = Θ(δ) + t 0 θ(δ, s) ds, where

63 Comparison with rational inattentiveness model Stopping time τ(p) & price gap p(t, p) for initial gap p P(t, δ) = Θ(δ) + t 0 θ(δ, s) ds, where invariant G(p) w/density: g (p 1,..., p n ) = f ( p p 2 n) adj. to surface ( n j=0 Θ(δ) = δ p ) j(0) dg (p (0)) n p(0) ιδ ȳ

64 Comparison with rational inattentiveness model Stopping time τ(p) & price gap p(t, p) for initial gap p P(t, δ) = Θ(δ) + t 0 θ(δ, s) ds, where invariant G(p) w/density: g (p 1,..., p n ) = f ( p p 2 n) adj. to surface ( n j=0 Θ(δ) = δ p ) j(0) dg (p (0)) n p(0) ιδ ȳ θ(δ, t) : density = derivative w.r.t. t of contribution to P for 0 < s t: P(t, δ) Θ(δ) = [ n j=0 E p j (τ(p), p) 1 {τ(p) t} p = p(0) ιδ ]dg (p (0)) n p(0) ιδ <ȳ back

65 Comparison with rational inattentiveness model Keep N ( p i ) fixed as we compare economies with different Kur( p) M n,l (δ; N ( p i ), Std[ p i ]) δ ɛ Kur ( p i ) 6 N ( p i ) effect as ratio to Calvo s effect.

66 Comparison with rational inattentiveness model Keep N ( p i ) fixed as we compare economies with different Kur( p) M n,l (δ; N ( p i ), Std[ p i ]) δ ɛ Kur ( p i ) 6 N ( p i ) effect as ratio to Calvo s effect. Special cases: n = 1, l = 0: Golosov-Lucas, Kur ( p i ) = 1 n = 1, l (0, 1): Calvo-Plus (NS), Kur ( p i ) (1, 6) n = 2, l (0, 1): Midrigan, Kur ( p i ) (1.5, 6) n =, l = 0: Taylor or rational inattentiveness", Kur ( p i ) = 3 n, l 1: Calvo, Kur ( p i ) = 6 back to figure

67 Comparison with rational inattentiveness model The firm s value of information: log(z) BM w/drift Perfect information Max static profit Π (z t ) max p Π(p, z t ), with maximizer p t = η η 1 (1/z t) Expected profit t periods ahead, assuming p t = pt, grows at rate b E 0 [ Π(p t, z t ) ] = Π (z 0 ) e b t

68 Comparison with rational inattentiveness model The firm s value of information: log(z) BM w/drift Perfect information Max static profit Π (z t ) max p Π(p, z t ), with maximizer p t = η η 1 (1/z t) Expected profit t periods ahead, assuming p t = pt, grows at rate b Imperfect information Max expected static profit max p E 0 [ Π(p t, z t ) ] = Π (z 0 ) e b t E 0 [ Π(p, z t ) ], with maximizer ˆp t = η η 1 E 0[1/z t ] Expected profit t periods ahead, assuming p t = ˆp t, grows at rate a E 0 [ Π(ˆp t, z t ) ] = Π (z 0 ) e a t The value of information: b a = η (η 1) σ 2 /2

69 Comparison with rational inattentiveness model Information time-line: current observation date: pay observation cost θ Π (z) learn production cost z learn signal ζ on future obs. cost τ periods later: learn z L( ; τ z) draw θ F( ; τ ζ) draw ζ G( θ )

70 Comparison with rational inattentiveness model Information time-line: current observation date: pay observation cost θ Π (z) learn production cost z learn signal ζ on future obs. cost τ periods later: learn z L( ; τ z) draw θ F( ; τ ζ) draw ζ G( θ ) Firm value function: V (z, ζ) = Π (z) v(ζ) v(ζ) = max τ R + τ 0 + e ( ρ+b)τ [ Optimal policy τ(ζ) e ( ρ+a) t dt + θ ζ ζ v (ζ ) dg (ζ θ ) df (θ ; τ ζ) } {{ } expected continuation value θ ] θ df (θ ; τ ζ) } {{ } expected cost a, b growth rate of profits w/no (w/perfect) information: b a = η(η 1) σ2 2. back

71 Comparison with rational inattentiveness model Golosov-Lucas back to IRF back to 2 useful results menu costs and phillips curves 193 Fig. 7. Approximate (dashed lines) and exact (solid lines) impulse-responsefunctions: responses of output to a one-time increase (impulse) in the level of money. Initial levels are normalized to one. In this regression, we obtain the estimate b p.049 with the standard error.008. Thus an increase in nominal wage rates leads to an increase in real output, as in standard Phillips curve regressions, but the effect is very small. This conclusion is not sensitive to different specifications of the parameters (m, j m ). Solid line: fixed point on path of aggregate consumption. Dashed line: keeps aggregate consumption at steady state.

72 Quality of GE approximation Quality of GE Approximation GE feedback are small (second order) Numerically: e.g. Golosov-Lucas. Analytically: as in Yun-Rotemberg-Woodford and especially Gertler-Leahy. Verify (numerically) effects of all approxiamtions: Case of n = 1 product. Add inflation of 2 % (instead of zero) Only productivity shock, and death (substitution) of products. (instead of productivity + negatively correlated demand shock) CES demand (instead of 2nd order approximation) Exact ideal price index (instead of approximation). Same: scaling by Std( p) N( p i ) + shape + location of max Cum. IRF

73 NON linear GE model Effects in non-linear GE model (n = 1, µ = 0.02) Figure: Output response to a 1% monetary shock Calvo (Substitution Only) Menu Cost Only Caballero Reis (Observation Cost Only) Output impulse response in % deviations from steady state Months from the shock param (relative to 1,1,1): N ( p i ) = 1.3, Std( p) = 0.8, ɛ = 2 Std( p) N( p i ) ɛ = 0.3 Source: Alvarez-Lippi-Paciello (2012) so

74 NON linear GE model Real effects in non-linear GE model (n = 1, µ = 0.02) Figure: Cumulated output effect Calvo (Substitutions Only) Menu Cost Only Caballero Reis (Observation Cost Only) 3.5 Cumulative "Output Effect" M( ), in % Monetary shock, in % param (relative to 1,1,1): N ( p i ) = 1.3, Std( p) = 0.8, ɛ = 2 Std( p) N( p i ) ɛ = 0.3 Source: Alvarez-Lippi-Paciello (2012) so

75 NON linear GE model Policy rules in non-linear GE model (n = 1, µ = 0.02) Figure: Threshold rules 9 8 Months elapsed frm the shock Price Gap: g t Source: Alvarez-Lippi-Paciello (2012) back to IRF back to 2 useful results

76 NON linear GE model Selection on size and time Compare the average price change of firms that adjust t periods after shock Golosov-Lucas: early on almost all adjustments upwards. graph back Selection on size decreases with l and n. Calvo: adjustments are independent of price gaps. Taylor: adjustments are independent of each price gaps. Any case with n = avg. price change δ every horizon t Difference due to distribution of times to adjust {τ}. In general, when there is no selection in size, M(δ) = δ [ ] 1 + CV (τ) ɛ 2 N ( p i ) So higher variability of times to adjustments {τ} increases area under IRF.

77 NON linear GE model Interpreting ψas information processing cost back Firms observes profit from n products, which are proportional to p 2. Firms don t know profits of each product line separately. If they pay ψ disentangle profits from each product, and can change prices accordingly. In this case ψ covers other activities than setting new prices.

78 NON linear GE model Results for Impulse Responses : scaling P n,l (δ, t): aggregate price level t periods after the monetary shock δ Scaling and Stretching: IRF P of economy with (Std[ p], N ( p i )) at (δ, t) is a scaled version of one for δ/std[ p] and stretched horizon N ( p i ) t: P n,l (δ, t; N ( p i ), Std[ p] ) ( ) δ = Std[ p] P n,l Std[ p], N ( p i) t ; 1, 1 back

79 NON linear GE model P n,l (δ, t): Response of CPI to shock δ = 1% Economy with n = 1 Economy with n = Price Level Response (in percentage) λ /Na=0.01 λ /Na=0.5 λ /Na=0.99 Price Level Response (in percentage) λ /Na=0.01 λ /Na=0.5 λ /Na= Months after the shock Months after the shock Scales depend on N ( p i ) = 2, Std( p) = 0.15 ; Shape on n, l back