Monetary Policy Implications of State-Dependent Prices and Wages

Monetary Policy Implications of State-Dependent Prices and Wages James Costain, Anton Nakov, Borja Petit Bank of Spain, ECB and CEPR, CEMFI The views expressed here are personal and do not necessarily coincide with the official views of Banco de España, ECB, or the Eurosystem. Philadelphia, January 28 Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 / 4

Motivation Nominal rigidity of some form is a key feature of most monetary models Common frameworks: Calvo (983), Rotemberg (982), Taylor (979) 2 Golosov-Lucas (27): state dependent (SD) model based on menu cost implies monetary shocks have trivial real effects The reason is endogenous selection : the most misaligned prices get reoptimized, so the price level is more flexible than a Calvo model implies 3 But newer SD pricing models deliver substantial money non-neutrality, closer to Calvo The reason is a weaker selection effect: Midrigan (2), Alvarez et al. (2), Matejka (2), Costain and Nakov (2, 25) 4 These new models match better retail price microdata, and respond well to big changes in the environment, e.g. VAT shocks (Karadi and Reiff 26) Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 2 / 4

Motivation 9 Unlike applied DSGEs, studies of state-dependent pricing mostly ignore all other frictions: sticky prices only Takahashi (27) is the only existing analysis of the interaction between SD sticky prices and SD sticky wages Takahashi ignores idiosyncratic shocks, so cannot match histograms of price or wage changes (the usual targets of the newer SD models) In this paper we compare model to price adjustment data and wage adjustment data simultaneously 2 We evaluate the role of both rigidities, simultaneously, for monetary policy 3 Huang and Liu (22) suggest that wage stickiness is more important than price stickiness for money non-neutrality Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 3 / 4

This paper Studies state dependent prices and wages simultaneously 2 Nominal rigidities following Logit Price Dynamics (Costain-Nakov, 25) Main assumption: precise decisions are costly 3 Game theoretic approach: control costs Postulate a cost function for precision Implies mistakes occur in equilibrium If precision is measured by entropy, then choices distributed as logit 4 Market structure following Erceg, Henderson, and Levin (2) Firms are monopolistic suppliers of goods, subject to a Calvo friction Workers are monopolistic suppliers of labor, subject to a Calvo friction 5 This paper: Erceg-Henderson-Levin (2) meets Costain-Nakov (25) Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 4 / 4

Model: monopolistic firms Profits: Firm i s demand: Yit = Y tp ɛ t P ɛ it Firm i s output: Yit = A it N it, where log A it is AR() Profits: Ut(P it, A it ) P it Y it W tn it Control variables: Firm adjusts its price Pit Current Pit remains in effect until firm sets a new price P Output and labor are demand driven. Frictions: Adjustment itself is costless (zero menu costs) But greater precision requires more decision time, so decisions are costly Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 5 / 4

Costs of decision-making: price choice Think of decisions as probability distributions over alternatives. Assume precision is costly. Let π(p) be a firm s chosen distribution over its log real price p. Assumption. The time cost τ of decision π is: κ π D(π η) κ π π(p) ln ( π(p) η(p) ) dp where η(p) is an exogenous default decision distribution. Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 6 / 4

Probability of price gaps Price choice Dirac delta distribution: maximum decision cost Logit distribution: intermediate decision cost Distance from optimal price Uniform distribution: zero decision cost Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 7 / 4

Costs of decision-making: timing choice Let λ be the probability of making a decision in the current period. Assumption 2. The time cost µ of choosing whether or not to make a decision is: ( ) ( κ λ D (λ, λ) ( λ, λ) κ λ λ log λ λ + ( λ) log λ ) λ where λ is an exogenous default probability. Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 8 / 4

Repricing probability Zero-one state-dependence: maximum decision cost Adjustment probability _ λ Smooth state dependence: intermediate cost Constant probability: zero decision cost Distance from optimal price Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 9 / 4

Bellman equations (real) Real value of producing at current firm-specific state (p, a): v t (p, a) = u t (p, a) + max [( λ)v et (p, a) + λṽ t (a) w t κ λ D ( (λ, λ) ( λ, λ) )] λ Where ṽt(a) is the firm s expected value, conditional on adjustment: ṽ t(a) = max π( p)v t e ( p, a)d p w tκ πd(π η) π( p) s.t. π( p)d p = And v e t (p, a) is the expected value, conditional on unchanged nominal price: } vt e (p, a) = E t {q t,t+v t+(p i t+, a ) a Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 / 4

Distribution of actions Both price distribution and probability of decision are weighted logits: Distribution of prices, conditional on decision: ( ) v e t η(p) exp (p,a) κ πw t π t (p a) = ( ) η( p) exp v e t ( p,a) κ πw t d p Probability of making a decision: λ t (p, a) = λ λ + ( λ) ( ), exp dt(p,a) κ λ w t Where d t (p, a) is the real loss from inaction: d t (p, a) = ṽ t (a) v e t (p, a) Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 / 4

Adding wage stickiness in an analogous way Next, do wage stickiness too Model wages and prices analogously, as in Erceg-Henderson-Levin (2) We assume each worker sells a distinct type of labor in a monopolistically competitive fashion to many firms So we are not yet addressing any other labor market frictions No search and matching, no unemployment Study effects of monetary shocks in a control cost model, assuming: Sticky prices and wages Sticky prices, flexible wages Flexible prices, sticky wages Flexible prices and wages And compare results to Calvo model Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 2 / 4

Model: monopolistic supply of labor Firm j s labor input is an aggregate of differentiated labor types i: N jt = { ɛn ɛn Nijt } ɛn ɛn di Worker i s effective labor N ijt is the product of labor time H ijt and worker-specific productivity Z it : N ijt = Z it H ijt, where log Z it is AR() Let W it be worker i s wage per unit of time, The aggregate wage index W t is: W t = { ( Wit Z it ) ɛn di} ɛn. Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 3 / 4

Model: monopolistic supply of labor Demand for labor time of worker i is: H it = H t (W it, Z it ) Z ɛn it N t W ɛn t W ɛn it. Households utility is: u(c t ) X (H t + µ w t + τ w t ) + ν(m t /P t ) where µ w t and τ w t are time devoted to wage decisions Then the marginal value of time is ξ t P t u (C t ) X (H t + µ w t + τ w t ) Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 4 / 4

Costs of decision-making Let π w (w) be a worker s chosen distribution over its log real wage w. Let ρ be the probability of making a decision in the current period. Assumption 3. The time cost τ w of decision π w is: κ w D(π w η w ) κ w π w (w) ln ( π w (w) η w (w) where η w (w) is an exogenous default decision. ) dw Assumption 4. The time cost µ w of choosing whether to make a decision is: ( ) ( κ w D (ρ, ρ) ( ρ, ρ) κ w ρ log where ρ is an exogenous default probability. ) ρ + ( ρ) log ρ ρ ρ Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 5 / 4

Bellman equation (real) l t (w, z) = max τ w,µ w,ρ,π w ( w) ew h t (w, z) X (h t(w, z) + τ w + µ w ) u (C t ) + ( ρ)lt e (w, z) + ρ π w ( w)l t e ( w, z)d w s.t. π w ( w)d w =, ( π ρκ w π w w ) ( w) ( w) ln η w d w = τ w, ( w) ( ) ( )] ρ ρ κ ρ [ρ ln + ( ρ) ln = µ w. ρ ρ Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 6 / 4

Distribution of actions Both wage distribution and probability of decision are weighted logits: Distribution of wages, conditional on decision: ( ) η w l e t (w) exp (w,z) πt w κ w ξ t (w z) = ( ) ηw (w ) exp l e t (w,z) κ w ξ t dw Probability of making a decision: ρ t (w, z) = ρ ρ + ( ρ) exp ( ), d w t (w,z) κ ρξ t Where d w t (w, z) is the real loss from inaction: d w t (w, z) = l t (z) l e t (w, z) Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 7 / 4

RESULTS: LINEAR LABOR DISUTILITY X (h) = χh Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 8 / 4

Common parameters (same in all specifications) Discount factor β 2 =.4 Golosov-Lucas (27) CRRA γ = 2 Ibid. Labor supply χ = 6 Ibid. MIUF coeff. ν = Ibid. Elast. subst. ɛ = 7 Ibid. Money growth µ 2 =.2 Dominick s dataset: 2% annual inflation Shocks to firms Persistence prod. ρ =.95 Blundell-Bond (2) Std. dev. prod. σ =.6 Eichenbaum et. al. (29) Shocks to workers Persistence prod. ρ =.95 Same as firms Std. dev. prod. σ =.6 Same as firms Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 9 / 4

Versions compared We compare six calibrations of the model: V: Benchmark. Sticky prices and wages: κπ = κ λ = κ w = κ ρ =.7 V2: Semi-flexible prices and sticky wages: κπ = κ λ =.7 V3: Flexible prices and sticky wages: κπ = κ λ =.7 V4: Sticky prices and semi-flexible wages: κw = κ ρ =.7 V5: Sticky prices and flexible wages: κw = κ ρ =.7 V6: Flexible prices and flexible wages: κπ = κ λ = κ w = κ ρ =.7 Note: This is the estimate of the benchmark model in Logit price dynamics. We will also compare each version to a Calvo model with sticky prices and wages with the same frequency of adjustment. Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 2 / 4

Density Density Density Density Density Density Nonzero price and wage changes: varying stickiness Nonzero price and wage changes: varying decision cost.25 Sticky prices and wages.5 Sticky prices and wages.2.5..5 -.5.5 Log price changes Semi-flexible prices and sticky wages.25.2.5..5 -.5.5 Log price changes Almost flexible prices and sticky wages.25.2.5..5 -.5.5 Log price changes.4.3.2. -.5.5 Log wage changes Sticky prices and semi-flexible wages.5.4.3.2. -.5.5 Log wage changes Sticky prices and almost flexible wages.5.4.3.2. -.5.5 Log wage changes Costain/Nakov/Petit Logit prices / wages June 26 26 / Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 2 / 4

Price.2.5..2.5 Cost Price Firms: Logit probabilities Cost Firms: Probability of adjustment.5.2. -.2 -.2 Price.5.2 Cost Probability Probability Price and wage setting: sticky prices and wages (V).5.8.6.2.4.5 Workers: Logit probabilities..5.3.2. -.5 Probability.2 -.5 Probability -.2 Price.2 Cost Firms: Probability of adjustment.5price as log deviation from flexible optimum.5 Price as log deviation from flexible optimum.2 Workers: Logit probabilities -.2.8.2.6.4.2 -.5 -.2.2 Productivity -.2.8.6.4.2.8.4..2.3.4.3Wage as log deviation from flexible optimum.2 Costain/Nakov/Petit Productivity -.2 -.2 Wage.2 Productivity.2 Workers: Probability of adjustment Probability -.2 Workers: Probability of adjustment.6.2 Workers: Logit probabilities -..5 Wage Probability -.2 Price as log deviation from flexible optimum.2 Workers: Probability of adjustment Workers: Logit probabilities.4 Productivity. Wage Workers: Probability of adjustment.6 -.2 Wage.3.2..3.2.2 -.5.5.8Price as log deviation from flexible optimum.2.2 Probability -.2.2 Firms: Logit probabilities Probability -...2.3.4.8Wage as log deviation from flexible optimum.6 Logit prices /.4 wages. -...2.3.4 Wage as log deviation from flexible optimum Costain-Nakov-Petit June 2.2 -...2.3.4 Wage as log deviation from flexible optimum State-dependent prices and wages Philadelphia, January 28 22 / 4

Steady-state behavior and decision costs V V3 V5 V6 Both sticky Fl-P, St-W St-P, Fl-W Both flex. Frequency and size of adjustments (%): Price adj. freq.. 54.4.4 54.4 Wage adj. freq. 6.2 6.4 7.28 6.95 Abs( ln p) 8.57 4.76 8.57 4.76 Abs( ln w) 6.4 6.6.98 2.29 Costs as % of revenues: Price setting costs.5.7.5.7 Price timing costs.37.3.37.3 Loss w.r.t. full rationality.78.3.78.3 Wage setting costs.3.4.4.4 Wage timing costs.4.5.4.3 Loss w.r.t. full rationality.62.7.8.7 Note: Firms costs stated as percentage of average revenue. Workers costs stated as percentage of average labor income. Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 23 / 4

Money growth Labor Price inflation Wage inflation Costain-Nakov-Petit Costain/Nakov/Petit State-dependent Logit prices prices / wages and wages Philadelphia, January June 28 26 283 / / 457 Consumption Real wage Money supply shock: effectsof of price and wage stickiness V: V: sticky, sticky, V3: V3: Pflex/Wsticky, Pflex/Wsticky, V5: V5: Psticky/Wflex, Psticky/Wflex, V6: V6: flexible flexible 4 3.8.6.4 V V3 V5 V6 3 2 2.2 5 5 2-2 - 2 3 4 3 2 3 2 2-5 5 2-2 - 2

Labor Wage inflation Real wage Money growth Price inflation Consumption Money Moneysupply supplyshock: shock: effects effectsofof stickiness stickiness (Calvo (Calvomodel) V: V: sticky, sticky, V3: V3: Pflex/Wsticky, V5: V5: Psticky/Wflex, V6: V6: flexible flexible.8.6.4 V V3 V5 V6.4.3.2 Calvo, linear disutility 2.5 2.5.2..5 2 3 4 2 3 4 2 3 4 2.5.4.8 2.3.6.5.5.2..4.2 2 3 4 2 3 4 2 3 4 Costain/Nakov/Petit Logit prices / wages June 26 33 / Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 29 / 4

Main findings: linear case Decreased decision costs for P or W have the expected effects: Make adjustment more frequent Make average adjustment smaller Decrease time devoted to the decision 2 Sticky wages generate more nonneutrality than sticky prices If W is flexible, stimulative effect of money supply increase is offset by W P Model with sticky wages and flexible prices generates most of the nonneutrality observed in the model in which both are sticky 3 Control costs on P and W recovers roughly half of the nonneutrality observed in an analogous Calvo model Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 3 / 4

RESULTS: CONVEX LABOR DISUTILITY X (h) = χ +ζ h+ζ, ζ =.5 Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 3 / 4

Density Density Density Density Density Density Figure 6: Distribution of nonzero price and wage changes: varying stickiness (ζ =.5). Nonzero price and wage changes: varying decision cost.25 Sticky prices and wages.4 Sticky prices and wages.2.5..5.3.2. -.5.5 Log price changes Semi-flexible prices and sticky wages.25 -.5.5 Log wage changes Sticky prices and semi-flexible wages.4.2.5..5.3.2. -.5.5 Log price changes Almost flexible prices and sticky wages.25 -.5.5 Log wage changes Sticky prices and almost flexible wages.4.2.5..5.3.2. -.5.5 -.5.5 Log price changes Log wage changes Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 32 / 4

Price and wage setting: sticky prices and wages (V) Notes: Distribution of adjustments and adjustment probability for prices (top four panels) and wages (bottom four panels) under nonlinear labor disutility (ζ =.5). Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 33 / 4

Steady-state behavior and decision costs VN V3N V5N V6N Both sticky Fl-P, St-W St-P, Fl-W Both flex. Frequency and size of adjustments (%): Price adj. freq. 7.74 49.6 7.78 5.4 Wage adj. freq. 7.44 8.34 22. 22. Abs( ln p) 7.3 4.2 7.24 3.99 Abs( ln w) 3.26 2.85 3.85 3.85 Costs as % of revenues: Price setting costs.49.7.47.6 Price timing costs.4.3.4.3 Loss w.r.t. full rationality.87.5.83.5 Wage setting costs.96.8.3.3 Wage timing costs.67.77.. Loss w.r.t. full rationality 4.7 4.5.3.3 Note: Firms costs stated as percentage of average revenue. Workers costs stated as percentage of average labor income. Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 34 / 4

Labor Wage inflation Real wage Money growth Price inflation Consumption Money supply shock: effects of price and wage stickiness V: sticky, V3: Pflex/Wsticky, V5: Psticky/Wflex, V6: flexible Figure 8: Money growth shock: effects of nominal rigidity. Error-prone pricing, ζ =.5. 2 2.5.8.6.4.2 V V3 V5 V6.5.5 2.5.5 5 5 2 3 2.5 5 5 2 2.5 2 -.5 5 5 2 2.5 2 2.5.5.5.5.5.5 5 5 2 -.5 5 5 2 -.5 5 5 2 Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 39 / 4

Labor Wage inflation Real wage Money growth Price inflation Consumption Money supply shock: effects of stickiness (Calvo model) V: sticky, V3: Pflex/Wsticky, V5: Psticky/Wflex, V6: flexible Figure 9: Money growth shock: effects of nominal rigidity. Calvo pricing, ζ =.5..6 2.5.8.6 V V3 V5 V6.4 2.5.4.2.2.5 2 4 2.5 2 4.8 2 4.5 2.6.5.4.2.5.5 2 4 -.2 2 4 2 4 Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 4 / 4

Conclusions We study a DSGE model with SD prices and SD wages 2 Combines monopolistic competition in goods and labor inputs, following Erceg, Henderson, and Levin (2), with nominal rigidity derived from costly decision-making, following Costain and Nakov (25) 3 First paper to study state dependence in prices and wages in a model with idiosyncratic shocks, for comparison to microdata 4 We find that wage stickiness is more likely to cause persistent effects of monetary shocks than price stickiness 5 Huang and Liu (22) reported the same finding for a time-dependent model; we are the first to study this issue in a state-dependent model 6 With nonlinear labor disutility, decreasing price stickiness, in the presence of sufficient wage stickiness, increases persistence of real effects of money shocks Costain-Nakov-Petit State-dependent prices and wages Philadelphia, January 28 4 / 4