Menu Costs and Phillips Curve by Mikhail Golosov and Robert Lucas. JPE (2007)

Menu Costs and Phillips Curve by Mikhail Golosov and Robert Lucas. JPE (2007) Virginia Olivella and Jose Ignacio Lopez October 2008

Motivation Menu costs and repricing decisions Micro foundation of sticky prices: menu costs models Two central issues for menu cost models 1 Firm s decision to reprice or not (timing of the updating) 2 Size of price changes Do Calvo models do a good job describing pricing behavior? 1 Repricing is more frequent in high-in ation environments (Calvo models assume constant repricing) 2 New data shows that prices change more than what usually assumed in Calvo models 3 Constant probability of price-adjustment rules out any selection e ect ( rms that change prices may be the ones whose prices are most

Motivation Modeling GL build a model where the rm s decision of reprice is along the two dimensions: size and time

Consumers Households Continuum of in nitely lived households each of which consumes a continuum of goods (Ω = PxV ) Z c t = C t (p) 1 Ω ε/ε 1 (1/ε) φ t (dp, dv)

Consumers Households Continuum of in nitely lived households each of which consumes a continuum of goods (Ω = PxV ) Z c t = C t (p) 1 Ω ε/ε 1 (1/ε) φ t (dp, dv) Households supplied labor (l) in a competitive market in exchange for a wage w and decide how much money to hold ˆm Real cash holdings provide utility to households. The opportunity cost of cash holdings is R

Consumers Optimization problem Preferences Z Max Ct (),l t, ˆm t E 0 1 e ρt 1 γ c1 γ t ˆmt αl t + log dt P t

Consumers Optimization problem Preferences Z Max Ct (),l t, ˆm t E 0 1 e ρt 1 γ c1 γ t ˆmt αl t + log dt P t Life-time budget constraint. (multiplier λ) Z Z E Q t pc t (p) φ t (dp, dv) + R t ˆm t w t l t Π t dt m 0 0 Ω

Consumers First-order conditions e ρt c γ t c 1/ε t C t (p) 1/ε = λq t p (C t (p)) e ρt = λq t R t ( ˆm t ) ˆm t αe ρt = λq t w t (l t ) 1 Let s propose an equilibrium where R t is constant 2 w t inherits the stochastic properties of m t w t = αrm t 3 d log (m t ) = µdt + σ m dz m

Firms Firms The production function exhibits CRS and productivity follows a mean reverting stochastic process C t (p) = v t l t d log (v t ) = η log (v t ) d t + σ v dz v Prices can be adjusted any time but rms face a menu cost denominated in terms of labor units From the optimization problem of the households, rms face the following demand schedule αp ε C t (p) = c 1 w t t εγ

Firms Pricing strategy Price setting: Firms choose both a time length T for their prices and a optimal new price q. Present value of a rm ϕ (p, v, w, φ t ) φ t joint distribution of (p,v) across rms ϕ (p, v, w, φ t ) = 2 max E 6 t 4 T R t+t w Q t s C s (p) p s v s ds + ϕ q, vt+t, w Q T max t+t, φ t+t q kw t+t 3 7 5

Market clearing Market clearing Markets: money,labor, nal (consumption) good m t = ˆm t Z C t (p) l t = φ Ω v t (dp, dv) + kυ t αp ε C t (p) = v t l t = c 1 εγ where Υ t = Υ t is the number of repricing rms. w t t

Idiosyncratic shocks First case: Only idiosyncratic shocks Both the money supply and the nominal wage have a drift of µ and variance of zero (deterministic) Conjecture an equilibrium where the distribution across rms is invariant (φ t = φ) and then aggregate consumption is constant (c t = c) Using the scaling property, GL proposed a solution of the form where x = p/w and ϕ (p, v, w) = wψ (x, v) Z T ψ (x, v) = max E e ρt ( c) 1 T 0 +e ρt max x 0 ψ x 0, v (T ) k εγ (αx t ) ε 1 x t dt v t

Idiosyncratic shocks Fix-point solution 1 Conjecture c i 2 Solve for the value function ψ (x, v) 3 Find the policy function and the associate distribution of prices over the state of idiosyncratic shocks 4 Compute c i+1 = α 1 ε R x 1 ε φ t (dx, dv; c) 1/[1 γ(ε 1)] and update your initial guess. Fix point c i = c i+1

Idiosyncratic shocks Discretization and state approximation Approximate the problem by a discrete time version with increments of size h (for time) and state space S = X xv ψ (x, v) = Π (x, v) t + e max r t x 0,v 0 π(x 0, v 0 jx, v)ψ (x 0, v 0 ), max ξ Π (ξ, v) t + e r t x 0,v 0 π(x 0, v 0 jξ, v)ψ (x 0, v 0 ) k where Π (x, v) = ( c) 1 εγ (αx t ) ε x 1 v and π is a transition function de ne on SxS (Markov-chain probabilities)

Calibration Calibration parameter value target/description ρ discount rate 0.4 interest rates γ risk aversion 2 usual RBC ε elasticity of sub. 7 markups of 16% α disutility labor 6 L=37% µ drift in ation process.0064 quarterly in ation rate k menu cost.0025 frequency of price changes n productivity shock.55 log-price increase over new prices σ 2 v variance prd. shock.011 standard deviation new prices

Qualitative features Qualitative features of this case

Results Results

Aggregate shocks Adding aggregate shocks One-time jump of money growth rate 1 Jump from µ to µ(1 + h) leads to a jump of wages by the same amount. Output increases at most by (1 + h) 1/γ 2 The fraction of rms that reprice is higher. Firms that reprice are the ones that have the most incentives to reprice (change in prices is large)

Aggregate shocks Self-selection: comparison between Calvo and GL

Aggregate shocks Hazard functions Absent of aggregate shocks, the model predicts a downward slopping hazard function. Recent updated prices are most likely to come from high-productive rms (high-sensitive to price misalignments) and have high probability to change again in the near future. Old prices most likely come from low-productive rms (with large inaction bands) and are unlikely to change.( atter curve) Elasticity of substitution: the larger the elasticity the less is the di erence in the inaction band between high and low-productive rms If aggregate shocks are present the model predicts upward slopping hazard function, provided aggregate shocks are persistent and idiosyncratic innovations fade out and have relative small variances.

Aggregate shocks Macro implications One time shock

Aggregate shocks Macro implications II Two aggregate shocks Monetary uctuations can account for less than 10% of the observed uctuations in output ( simulations of 40 quarters of data) Using the simulated series, GL run the following regression: log yt Q h = α + β log wt Q i log w Q t 1 The estimation of the parameters suggests that the Phillips-Curve holds ( ˆβ is 0.049) but the e ect is quite small

New features Three "New" Features of the Distribution of Price Changes Using scanner price data collected in retail stores and excluding sales, Midrigan documents: 1 Large number of small price changes 2 Fat Tails (excess Kurtosis) 3 Adjustment in tandem within a store =)Extending a standard menu cost model to a multi-product setting with economies of scope in technology of price adjustment can replicate these facts and generate larger aggregate uctuations than standard menu cost economies

Modeling di erences Multiproduct Menu Cost Model Two main departures from GL: 1 Multiproduct rms (vs single product rms) that face a xed cost of changing its entire menu of prices but, conditional on paying this cost, zero marginal cost of resetting any given price on the menu 2 Leptokurtic shocks (vs Gaussian shocks) =)These two features determine that the selection e ect present in GL is much weaker in here and thus responsiveness of aggregate price levels to monetary shocks is reduced generating larger aggregate uctuations

Quantitative results Quantitative Results: Parameters Preferences (Hansen 1985) U(c, n) = log(c) ψn Parameters (Benchmark)

Calibration Quantitative Results: Calibration Targets Calibration Targets

Calibration Role of Fixed Cost of Changing Entire Menu of Prices Small price changes will arise in equilibrium whenever at least one of the rm s two prices is hit by a su ciently large shock Firms are more willing to adjust prices in periods when their technology is higher (GL)

Calibration Matching Micro Features of the Data

Aggregate Implications Aggregate Implications Interactions in the costs of price adjustment + leptokurtic shocks are thus a necessary ingredient of a model capable of reproducing the microeconomic evidence and generating sizable business cycle uctuations from monetary disturbances

Aggregate Implications Real E ects from Money Shocks: MP vs GL Impulse Response to 1% increase in money growth rate

Aggregate Implications Real E ects from Money Shocks: MP vs GL Aggregate price level response (π): π = log P P 1 = fraction of adjusters {z } Extensive Margin mean price cond. on a price {z } Intensive Margin

Aggregate Implications Explaining the "Dampened" Price Response: Weaker Selection E ect Selection e ect in standard menu cost models (GL) Strength of selection e ect, density of desired price changes and adjustment hazards:

Aggregate Implications Explaining the "Dampened" Price Response: Selection E ect and Distribution of Price Changes E ect of money shock on the distribution of non-zero price changes

Aggregate Implications Counterfactuals 1 Calvo-Timing: Self-Selection and Synchronization are shut down 2 No Self-Selection =)Self Selection and Front Loading

conclusions Conclusions Standard single-product state-dependent pricing models are inconsistent with large number of small price changes and excess kurtosis of price changes found in the data These facts can be reconciled with state-dependent models if multi-product rms face interactions in the costs of adjusting This model can generate business cycle uctuations from nominal disturbances that are almost as large as in Calvo-style time-dependent models. A key feature of the calibration, the leptokurtic distribution of idiosyncratic disturbances, together with the assumption of economies of scale in the price adjustment technology, implies that the selection e ect that plays an important role in standard menu cost economies is much weaker in this setup