Price plans and the real effects of monetary policy

Transcription

1 Price plans and the real effects of monetary policy Fernando Alvarez and Francesco Lippi University of Chicago Einaudi Institute Rome (EIEF) Conference: Price Setting and Inflation, December 2015 Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

2 Introduction Many temporary price changes in micro data GOLDBERG & HELLERSTEIN LOCAL CURRENCY PRICE STABILITY 184 REVIEW OF ECONOMIC STUDIES FIGURE 1 Weekly retail and wholesale prices for Bass Ale. Prices are for a single six-pack and are from Zone observations. Source: Dominick s FIGURE 2 Weekly retail and wholesale prices for Corona. Prices are for a single six-pack and are from Zone observations. Source: Dominick s Should we count" the transitory price changes or ignore them? Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

3 Introduction Many temporary price changes in micro data GOLDBERG & HELLERSTEIN LOCAL CURRENCY PRICE STABILITY 184 REVIEW OF ECONOMIC STUDIES FIGURE 1 Weekly retail and wholesale prices for Bass Ale. Prices are for a single six-pack and are from Zone observations. Source: Dominick s FIGURE 2 Weekly retail and wholesale prices for Corona. Prices are for a single six-pack and are from Zone observations. Source: Dominick s Should we count" the transitory price changes or ignore them? This paper: build a model w/ temporary price changes matching the data compare the effect of monetary shocks w/ and w/o transitory price changes Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

4 Introduction do temporary price changes matter for macro? We study this question using an analytical model based on Eichenbaum, Jaimovich, Rebelo (2011) Problem setup: Firms track a desired profit maximizing price p (t) they can post a price from the current Price Plan P = {p L, p H } Price changes within the plan p L p H are costless Changing the Plan P requires paying a fixed cost Objective is to characterize: optimal plan, stopping times, aggregation, shock-propagation Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

5 Main results Introduction Analytics of the firm s choices (make complex problem tractable) Decreasing Hazard rate price of changes Plans increase price flexibility (reduce output effects of monetary shocks) - differs from Guimares-Sheedy, Eichenbaum-et al., Kehoe-Midrigan Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

6 General Equilibrium Set Up General Equilibrium Set Up GE setup: Woodford, Midrigan, Golosov-Lucas model with 2-price Plan Lifetime Utility : Z 1 CES aggregate : c(t) = 0 c(t) e r t Z 1 0 c k (t) 1 dk! 1 `(t) + log M(t) dt P(t) Linear technology c k (t) =`k(t) / Z k (t) and Z k (t) = exp ( Z k (t)). Equilibrium: constant nominal interest rate & wages W (t) =a M(t). Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

7 General Equilibrium Set Up General Equilibrium Set Up GE setup: Woodford, Midrigan, Golosov-Lucas model with 2-price Plan Lifetime Utility : Z 1 CES aggregate : c(t) = 0 c(t) e r t Z 1 0 c k (t) 1 dk! 1 `(t) + log M(t) dt P(t) Linear technology c k (t) =`k(t) / Z k (t) and Z k (t) = exp ( Z k (t)). Equilibrium: constant nominal interest rate & wages W (t) =a M(t). firm s profits : (p p, c) B (p p ) 2 where B = ( 1) 2 details ` units of labor paid to change the Price Plan Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

8 The firm problem firm problem process for desired price: dp (t) = dz(t) where shock volatility Value function: V (p, p L, p H )= min E { i,p i } 1 i=1 " 1 X i=1 Z i e rt min B (p(t) p (t)) 2 dt i 1 p(t)2p i 1 # 1X + e r i p = p (0), P 0 = {p L, p H } i=1 Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

9 The firm problem firm problem process for desired price: dp (t) = dz(t) where shock volatility Value function: V (p, p L, p H )= min E { i,p i } 1 i=1 " 1 X i=1 Z i e rt min B (p(t) p (t)) 2 dt i 1 p(t)2p i 1 # 1X + e r i p = p (0), P 0 = {p L, p H } i=1 Equivalent value function representation v(g ; g) V (p, p 0 g, p 0 + g) where g(t) p (t) p ( i ) Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

10 firm problem benefits of a 2-price plan flow cost: B min {(g g) 2, {(g + g) 2 } Period losses with Plans Period losses w/o Plans g Normalized desired price g HJB equation: rv(g; g) =B g 2 + g 2 2 g g v 00 (g; g) Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

11 firm problem Example of a policy: p(t) 2{p L, p H }, p (t) Notice changes of Plans vs changes within Plans Price price: p = p i ± g desired price p desired price at stopping times p i barrier for change of plan: p i ± ḡ time in years Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

12 firm problem Characterization of the Optimal policy: ḡ, g HJB equations rv(g; g) =B g 2 + g 2 2 g g v 00 (g; g) (1) v(g; g) apple + min g 0 v(0; g 0 ) optimality conditions v(ḡ; g) = + v(0, g), value matching (2) v 0 (ḡ; g) =0, smooth pasting v(0; g) =0, optimality of g g Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

13 firm problem Optimal decision rules: g, ḡ (Prop 1 and Prop 2) Optimal prices within a plan (recall p(t) =p i ± g ) rḡ 2 g = ḡ 2 ḡ 1 as r! 0 3 Optimal width of the plan ( adjust when g p (t) p i > ḡ ) ḡ = f r B, r B as r! 0 Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

14 Cross section predictions Frequency of price changes: model vs data Total (expected) number of price changes N = N T N T depends on model period : {z} temp. + N p {z} plan ( if! 0 then N!1 if > 0 then N p N p / where N p = 2 ḡ 2 Reference Price (EJR) = mode of prices within a period (say a quarter) Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

15 Cross section predictions Frequency of price changes: model vs data Total (expected) number of price changes N = N T N T depends on model period : {z} temp. + N p {z} plan ( if! 0 then N!1 if > 0 then N p N p / where N p = 2 ḡ 2 Reference Price (EJR) = mode of prices within a period (say a quarter) Data: Prices spend large fraction of time at reference price ; N large Model with N p > 0 (but small) and > 0: prices spend almost all the time at the reference price and N!1. Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

16 Cross section predictions Summary statistics on price setting behavior (weekly model = 1 52 ) # Price changes per year (N = N p + N w ) # Plan changes per year (N p ) # Reference Price changes per year Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

17 Cross section predictions Summary statistics on price setting behavior (weekly model = 1 52 ) # Price changes per year (N = N p + N w ) # Plan changes per year (N p ) # Reference Price changes per year Freq. of Price changes (per week) Freq. of Reference price change (per quarter) Fraction of time spent at reference price Fraction of time spent below the reference price reference price modal price within a quarter fraction time spent ref. price: EJR : 0.62, Dominick s: 0.77 Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

18 Cross section predictions Hazard Rate of Price changes Hazard: probability (per unit of time) of price change as function of duration Hazard rate of price changes Hazard for all price changes h(t) with lim t!0 h(t) t = 1 2 Hazard for plan changes h p (t) Model All price changes Only plan changes 1/2t N p π 2 2 π N 2 0 p time elapsed since adjustment, as fraction of Expected duration Percentage Points or h(t) 1 2 t Data (Campbell and Eden) EVIDENCE FROM U.S. SCANNER DATA Hazard Rates of Original Price Series Hazard Rates of Series with Blip Prices Replace Hazard Rates of Series with Sale Prices Replace >12 Price Age in Weeks Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

19 Impulse response Impulse response analysis Start with invariant cross-section distribution of desired prices: h(g) Shock: permanent increase of money supply % (nominal wages ) Simplification: decision rules in transition same as in St. St. Z t Agg. price level IRF P(t, )= ( )+ (s, )ds 0 Cumulated output response M ( ) Z 1 0 Y (t, ) dt = 1 Z 1 0 [ P(t, )] dt Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

20 Impulse response Impulse responses for models with same ḡ, 2 Models have identical number of Plan changes N p = 2 /ḡ with Plans without Plans Output deviation from steady state time (years) Cumulated output response M ( ) R 1 0 Y (t, ) dt = 1 R 1 0 [ P (t, )] dt Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

21 Impulse response Mechanics of the impulse response: shift h(g) h(g) h(g + δ) ḡ g g ḡ Normalized desired price g Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

22 Impulse response Why this difference? Non-negligible impact effect! A mass of firms increase prices from g to g on impact! on impact the price level jumps by 2/3 of Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

23 Impulse response Output effect following a small shock (analytics) For small shocks we use M( )= M 0 (0)+o( ) Cumulated output effect in menu cost model without plans M 0 MC (0) = ḡ2 6 2 = 1 6 N Cumulated output effect in menu cost model with plans M 0 (0) = ḡ = 1 18 N p Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

24 Impulse response Application of main result N: number of price changes, N p : number of PLAN changes M is approximate cumulated output after small shock W/ Price Plans W/out Price Plans Ratio With Plans Without Plans Theory M( )= N p M( )= 6 1 N M( ) M( ) = N 3 N p Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

25 Impulse response Application of main result N: number of price changes, N p : number of PLAN changes M is approximate cumulated output after small shock W/ Price Plans W/out Price Plans Ratio With Plans Without Plans Theory M( )= N p M( )= 6 1 N M( ) M( ) = N 3 N p Calibration weekly model =1/ >< M( )= >< 1 N p = 3.33, N = 15 M( )= 6 10 >: M( )= >: M( ) M( ) = 1 3 M( ) M( ) = Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

26 Relation to Kehoe-Midrigan Relation to Kehoe-Midrigan (2015); 3 parameters Assume exogenous arrivals of price-change opportunities, of 2 types: Permanent price changes, with arrival rate P Temporary price changes, with duration T, with arrival rate T Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

27 Relation to Kehoe-Midrigan Relation to Kehoe-Midrigan (2015); 3 parameters Assume exogenous arrivals of price-change opportunities, of 2 types: Permanent price changes, with arrival rate P Temporary price changes, with duration T, with arrival rate T Implies (recall: standard Calvo has T = 0): Total Number of adjustments per period: N = P + T Cumulated output effect 1 + e P T M( ) 1 P + T ( 1 e P T ) Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

28 Relation to Kehoe-Midrigan Relation to Kehoe-Midrigan (2015); 3 parameters Assume exogenous arrivals of price-change opportunities, of 2 types: Permanent price changes, with arrival rate P Temporary price changes, with duration T, with arrival rate T Implies (recall: standard Calvo has T = 0): Total Number of adjustments per period: N = P + T Cumulated output effect 1 + e P T M( ) 1 P + T ( 1 e P T ) As T! 0: real effects unchanged, but N can diverge In K-M: T = 1 12 and P T 0.9 =) e P T = Bottomline: N triples while M basically unchanged Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

29 Relation to Kehoe-Midrigan summary: what explains the smaller real effects? Flexibility does not come from changes in plans number of firms that change plans following a shock same as GL EJR has more flexibility (state dependence) within a Plan Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

30 Relation to Kehoe-Midrigan summary: what explains the smaller real effects? Flexibility does not come from changes in plans number of firms that change plans following a shock same as GL EJR has more flexibility (state dependence) within a Plan In particular: how do firms react to higher cost? - More firms will be at the high value of their plan. - Food for empirical work! do temporary p respond to shocks? Anderson, Nakamura et al (2015) vs Kryvstov and Vincent (2014) Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

31 evidence cyclical sales Cyclicality of Sales (Kryvstov Vincent, 2015) Frequency (solid) Sales BLS (weighted) 12 month centered moving avg Unemp. rate (dashed) Frequency (solid) Sales and substitutions Vavra (2014) 12 month centered moving avg Unemp. rate (dashed) Figure 5: Frequency of sales in the U.S. CPI data and unemployment rate Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

32 evidence cyclical sales Cyclicality of Sales (ANMSS, 2015) 7.5% 7.0% 6.5% 6.0% 5.5% 5.0% 4.5% Figure 6 Frequency of Sales and Unemployment Rate 9.5% 8.5% 7.5% 6.5% 5.5% 4.5% 4.0% 3.5% Mar 89 Mar 94 Mar 99 Mar 04 Mar 09 Mar 14 Sales Frequency (left axis) Unemployment Rate (right axis) Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

33 Concluding remarks Conclusions price-setting problem in a tractable GE set-up temporary price changes dampen real effects of monetary shocks state dependence (within-plan selection) boosts price flexibility Analogue results occur with exogenous adjustment times for Plans details Analogue results occur in model with transitory (iid) shocks details Bonus: decreasing hazard functions and hump-shaped IRFs Alvarez and Lippi (UofC, EIEF) Sticky plans BdF, Dec / 24

34 Extension 1: Calvo plans Exponentially distributed plan adjustments ( Calvo ) Firm maximizes profits given exogenous adjustment times: i Namely: firm allowed to reset PLAN: {p L, p H } with rate

35 Extension 1: Calvo plans Exponentially distributed plan adjustments ( Calvo ) Firm maximizes profits given exogenous adjustment times: i Namely: firm allowed to reset PLAN: {p L, p H } with rate Model yields the optimal Plan prices upon reset: p i ± g where g = p 2( + r)

36 Extension 1: Calvo plans Exponentially distributed plan adjustments ( Calvo ) Firm maximizes profits given exogenous adjustment times: i Namely: firm allowed to reset PLAN: {p L, p H } with rate Model yields the optimal Plan prices upon reset: p i ± g where g = p 2( + r) Invariant distribution of desired prices: h(g) = p 2 2 e p 2 g for all g

37 Extension 1: Calvo plans Impulse response function w/ exponential adjustments

38 Extension 1: Calvo plans Cumulated output after a small money shock For small shocks we use M( )= M 0 (0)+o( ) Cumulated output effect in Calvo model without plans M 0 (0) = 1 N Cumulated output effect in Calvo model with plans M 0 (0) = 1 2N p

39 Extension 2: iid shocks A model with iid shocks to desired prices Assume p (desired price) is iid with CDF F( ) Z v(p L, p H )= min x min B(pj x) 2 + v(p L, p H ), p j 2P + min {p 0 L,p0 H } B(ˆp x)2 + v(p 0 L, p 0 H) df (x)

40 Extension 2: iid shocks A model with iid shocks to desired prices Assume p (desired price) is iid with CDF F( ) Z v(p L, p H )= min x min B(pj x) 2 + v(p L, p H ), p j 2P + min {p 0 L,p0 H } B(ˆp x)2 + v(p 0 L, p 0 H) df (x) Simulation result calibrating 2 economies to the same freq. PLAN changes optimal reset PLAN as function of shock x 45 deg. line return point: p0 return point: p Response of output following a monetary shock with Plans without Plans optimal reset PLAN (P0,P1) Output response desired price: x model periods after the shock

41 Extension 2: iid shocks Effects of Price-Plans with iid shocks Calibration W/ Price Plans W/out Price Plans Ratio for both # N p = 11 N = 31, N ref = 3.4 N = 11, N ref = 2.8 With Plans Without Plans M( ) M( ) = 1 7 N ref = 1 N = 18, N p = 0.33 N = 3, N p = 3 M( ) M( ) = 1 2 Frequencies per year ; weekly model =1/52 Legend: freq. all prices N, freq. plans N p, freq. reference prices N ref back

42 Extension 2: iid shocks Hazards rates and monetary shocks Notice the difference: EJR has decreasing hazard rates (consequence of a selection effect ) KM has constant hazards (no selection effect) EJR-plans reduce real effects of monetary shocks more than KM-plans the reason is selection (state dependent) nature of adjustments Key margin of adjustment is the frequency, not the size of p consistent with evidence in Krystov and Vincent (2014)

43 details on firm problem Approximate profit function define the (log) desired price pt log W t Z t 1 define the price gap x p(t) p (t) (log prices) Discounted nominal profit product i as function of x (x, c(t)) / W (t) e rt c(t) 1 e x apple e x (0, where c(t) aggregate consumption (0, firm s profits : (x, c(t)) B (p(t) p (t)) 2 where B = ( 1) 2 back to GE Example in GL

44 details on firm problem Bellman equation ( 3 parameters: /B,,r ) applez V (p, p L, p H )=min E 0 e rt min B (p(t) p(t)2{p L,p H } p (t)) 2 dt + e r [ + V (p ( )) ] p = p (0) # where V (p ) min ( p L, p H )2R 2 V p, p L, p H = V, 8 p and dp (t) = dz(t) Invariance to translations e.g. a > 0 V (p, p L, p H )=V ( p + a, p L + a, p H + a) for all a, p, p L, p H

45 details on firm problem Redefine state space and value function The optimal price plans are centered and symmetric (Lemma 1): P i = { p i g, p i + g } where p i p ( i ) Redefine the state-space (shrink it by one dimension) v(g ; g) V ( p, p i g, p i + g) where g(t) p (t) p i The new state g is the normalized desired price

46 details on firm problem Characterization of cumulated output effect Expected cumulated output by a firm with g(t) =p (t) p ( i ) m(g) =E 4 Z 0 B C g(t) g sign(g(t)) A dt g(0) =g5 {z } y t where dg = dz, m(0) =0, m( g) = m(g) 3

47 details on firm problem Characterization of cumulated output effect Expected cumulated output by a firm with g(t) =p (t) p ( i ) m(g) =E 4 Z 0 B C g(t) g sign(g(t)) A dt g(0) =g5 {z } y t where dg = dz, m(0) =0, m( g) = m(g) 3 Cumulated output! M( ) = Z ḡ ḡ m(g + ) h(g) dg

48 details on firm problem Characterization of cumulated output effect Expected cumulated output by a firm with g(t) =p (t) p ( i ) m(g) =E 4 Z 0 B C g(t) g sign(g(t)) A dt g(0) =g5 {z } y t where dg = dz, m(0) =0, m( g) = m(g) 3 Cumulated output! M( ) = Z ḡ ḡ m(g + ) h(g) dg Useful because we have an ODE for m(g) : Example w menu cost: 0 = g + m 00 (g) 2 with boundary conditions m(0) =m(ḡ) =0 and known h(g) 2

49 details on firm problem Cumulated output effect with plans Expected cumulated output by a firm with g(t) =p (t) p ( i ) m(g) =E 4 Z 0 B C g(t) g sign(g(t)) A dt g(0) =g5 {z } y t where dg = dz, m(0) =0, m( g) = m(g) 3

50 details on firm problem Cumulated output effect with plans Expected cumulated output by a firm with g(t) =p (t) p ( i ) m(g) =E 4 Z 0 B C g(t) g sign(g(t)) A dt g(0) =g5 {z } y t where dg = dz, m(0) =0, m( g) = m(g) 3 Cumulated output! M( ) = Z ḡ ḡ m(g + ) h(g) dg Useful because we have an ODE for m(g) : Example (w/ plan): menu cost: 0 = g g sign(g)+m 00 2 (g) 2 with boundary conditions m(0) =m(ḡ) =0 and known h(g)

51 Inflation Lack of sensitivity to Inflation µ Inflation has only second order effect around µ = 0 on entire hazard rate h(t; µ) µ=0 = 0 Cumulated output: M 0 (0; µ) ; 0 ( µ=0 = 0 Main ideas: by symmetry h(t; µ) =h(t; µ) differentiating w.r.t. µ and evaluating at µ = 0 for any t by symmetry: M( ; µ) = M( ; µ) where M 0 ( ; ( differentiating w.r.t. µ and both sides and evaluating at (, µ) =(0, 0) back to hazard back to output effects

52 Inflation Compare Effects of a Monetary Shock : given parameters /B, 2, r W/ Price Plans W/out Price Plans Ratio With Plans Without Plans Effect M-shock M( )= N p M GL ( )= 6 1 N GL M( ) M GL ( ) = N GL 3 N p # plans/prices N p = 2 optimal threshold ḡ = ( ḡ ) 2 N GL = B ḡ GL = 2 ( ḡ GL ) B optimal # plans N p = p 18 /B N GL = p 6 /B Total Effect M( ) M GL ( ) = 1 p M and M GL approximations for small value of. ḡ and ḡ GL approximations for small value of product ( /B) r 2 2.

53 evidence cyclical sales Cyclicality of Sales (Kryvstov Vincent, 2015) Frequency (solid) Sales BLS (weighted) 12 month centered moving avg Unemp. rate (dashed) Frequency (solid) Sales and substitutions Vavra (2014) 12 month centered moving avg Unemp. rate (dashed) Figure 5: Frequency of sales in the U.S. CPI data and unemployment rate

54 evidence cyclical sales Cyclicality of Sales (ANMSS, 2015) 7.5% 7.0% 6.5% 6.0% 5.5% 5.0% 4.5% Figure 6 Frequency of Sales and Unemployment Rate 9.5% 8.5% 7.5% 6.5% 5.5% 4.5% 4.0% 3.5% Mar 89 Mar 94 Mar 99 Mar 04 Mar 09 Mar 14 Sales Frequency (left axis) Unemployment Rate (right axis)