Introduction. The Model Setup F.O.Cs Firms Decision. Constant Money Growth. Impulse Response Functions

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1 F.O.Cs

2 s and Phillips Curves Mikhail Golosov and Robert Lucas, JPE 2007 Sharif University of Technology September 20, 2017

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4 A model of monetary economy in which firms are subject to idiosyncratic productivity shocks as well as general inflation. Sellers can change a price only by incurring a real menu cost. The Caplin and Spulber example is unrealistic in too many respects to be implemented quantitatively This model is designed so that it can be realistically calibrated using a new data set on prices, assembled and described by Bils and Klenow (2004) and Klenow and Kryvtsov (2005)

5 The prediction of the calibrated model for the effects of high inflation on the frequency of price changes accords well with international evidence from various studies Also the main finding of the paper is that monetary shocks real effects are dramatically less persistent than in an otherwise comparable economy with time-dependent price adjustment

6 F.O.Cs The log of the money supply is m t is assumed to follow a Brownian motion with drift parameter µ and variance σ 2 m d log(m t ) = µdt + σ m dz m Where Z m denotes a standard Brownian motion with zero drift and unit variance There are also firm specific productivity shocks v t which are independent across firms: d log(v t ) = η log(v t )dt + σ v dz v Where Z v denotes a standard Brownian motion with zero drift and unit variance

7 F.O.Cs The state of the economy at date t includes the level of money supply m t and nominal wage rate w t The situation of an individual firm depends also on the price p t that it carries into t from earlier dates and its idiosyncratic productivity shock v t The state of the economy also depend on the distribution of firms φ t (p t, v t )

8 F.O.Cs At each date t, each household buys from every seller, and each seller is characterized by a pair (p, v), distributed according to a measure φ t (p t, v t ) The household chooses a buying strategy {C t (.)} where C t (p) is the number of units of consumption good that it buys from a seller who charges price p at date t [ c t ] ε/(ε 1) C t (p) 1 (1/ε) φ t (dp, dv) It also chooses a labor supply strategy {l t } and a money holding strategy { ˆm t }

9 F.O.Cs Consumer Preferences over time is: [ [ ( )] ] 1 E e ρt ˆmt 1 γ c1 γ t αl t + log dt 0 The consumer s budget constraint is: [ [ E Q t 0 ] ] pc t (p)φ t (dp, dv) + R t ˆm t W t l t Π t dt m 0 Π t : Profit income from holding of a fully diversified portfolio of claims on the individual firms plus any lump-sum transfers R t is the nominal interest rate and R t ˆm t represents the opportunity cost of holding cash P t

10 F.O.Cs F.O.Cs Money Holdings: e ρt 1 m t = λq t R t Labor-Consumption: e ρt c γ t c 1/ε t Labor Decision: e ρt α = λq t w t C t (p) 1/ε = λq t p It can be shown that there is an equilibrium in which: R t = R = ρ + µ w t = αrm t Thus log(w t ) follows the same path as log(m t ). Derivation of this equation depends on crucial assumptions about utility function

11 F.O.Cs If the firm leaves its price unchanged, its current profit level is: ( C t (p) p w ) t v t If it chooses any price q p its current profit level is: ( C t (q) q w ) t kw t v t Where the parameter k is the hours of labor needed to change the price, the real menu cost

12 F.O.Cs Let ϕ(p, v, w, φ t ) denote the present value of a firm. This firm chooses a shock-contingent repricing time T 0 and a shock-contingent price q to be chosen at t + T so as to solve: [ t+t ( ϕ(p, v, w, φ t ) = max E t Q s C s (p) p w ) s ds T t v s + Q T. max[ϕ(q, v t+t, w t+t, φ t+t ) kw t+t ] q ] (1)

13 F.O.Cs Using households F.O.Cs it is easy to show that: ( ) C t (p) = c 1 εγ αp ε t (2) w t w t Q t+s = e ρs (3) w t+s Using (2), (3), we can express the Bellman equation (1) as: [ t+t ϕ(p, v, w, φ t ) = max E t e ρ(s t) c 1 εγ t T t ( αp w s ) ε ( p w ) s ds + e ρt w. max[ϕ(q, v t+t, w t+t, φ t+t ) kw t+t ] w T q v s ]

14 We treat the special case in which the variance σ 2 m of the money growth and wage process is zero, so that the drift parameter µ is simply the constant rate of wage inflation In this situation, there is an invariant distribution φ for real prices x t = p w t and idiosyncratic shocks v, thus the consumption aggregate is: c t = [α 1 ε x 1 ε φt (dx, dv)] 1/[γ(ε 1)] = c

15 Then the Bellman equation can be written as: [ 1 t ( 1 εγ ϕ(wx, v, w) = max E e ρs c t (αx s ) ε x s 1 ) ds w T 0 v s ] + e ρt 1. max w [ϕ(w T x, v T, w T ) kw T ] T x Finally, the solution to the above equation is in the form of φ(p, v, w) = wψ(x, v) and can be studied with familiar methods

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17 The two bounds in the Figure 1 determine the region of inaction on which the firm s relative price x = p/w declines at the rate µ because of deterministic wage growth, and its productivity level v moves stochastically When the boundaries are reached the price is changed to the dotted line on the figure Note that getting prices right is more important when productivity shocks and hence quantities sold are high Next figure shows the necessity of including idiosyncratic shocks to describe the fraction of prices changed each month

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19 Responses to a one time increase in the level of money of %1.25

20 Responses to a one time increase in the level of money of %1.25

21 Impulse responses are much more transient than a standard time dependent model would predict

22 vs Calvo Pricing Comparing before and after distributions of individual prices to illustrate the reason for these different responses