menu costs and phillips curvesgsme.sharif.edu/~madanizadeh/files/macrophd/files/golosov-lucas.pdfif...

OutlineIntroductionThe Model

Constant Money GrowthImpulse Response Functions

Menu Costs and Phillips CurvesMikhail Golosov and Robert Lucas, JPE 2007

Sharif University of Technology

April 23, 2016

Special Topics in Macroeconomics Menu Cost



Introduction

The ModelSetupF.O.CsFirms’ Decision

Constant Money Growth

Impulse Response Functions




Introduction

I A model of monetary economy in which firms are subject toidiosyncratic productivity shocks as well as general inflation.Sellers can change a price only by incurring a real ”menucost”.

I The Caplin and Spulber example is unrealistic in too manyrespects to be implemented quantitatively

I This model is designed so that it can be realistically calibratedusing a new data set on prices, assembled and described byBils and Klenow (2004) and Klenow and Kryvtsov (2005)




Introduction

I The prediction of the calibrated model for the effects of highinflation on the frequency of price changes accords well withinternational evidence from various studies

I Also the main finding of the paper is that monetary shocks’real effects are dramatically less persistent than in anotherwise comparable economy with time-dependent priceadjustment




SetupF.O.CsFirms’ Decision

Setup

I The log of the money supply is mt is assumed to follow aBrownian motion with drift parameter µ and variance σ2m

d log(mt) = µdt+ σmdZm

Where Zm denotes a standard Brownian motion with zerodrift and unit variance

I There are also firm specific productivity shocks vt which areindependent across firms:

d log(vt) = −η log(vt)dt+ σvdZv

Where Zv denotes a standard Brownian motion with zero driftand unit variance





Setup

I The state of the economy at date t includes the level ofmoney supply mt and nominal wage rate wt

I The situation of an individual firm depends also on the pricept that it carries into t from earlier dates and its idiosyncraticproductivity shock vt

I The state of the economy also depend on the distribution offirms φt(pt, vt)





Setup

I At each date t, each household buys from every seller, andeach seller is characterized by a pair (p, v), distributedaccording to a measure φt(pt, vt)

I The household chooses a buying strategy {Ct(.)} where Ct(p)is the number of units of consumption good that it buys froma seller who charges price p at date t

ct ≡[∫

Ct(p)1−(1/ε)φt(dp, dv)

]ε/(ε−1)I It also chooses a labor supply strategy {lt} and a money

holding strategy {m̂t}





Setup

I Consumer Preferences over time is:

E

[∫ ∞0

e−ρt[

1

1− γc1−γt − αlt + log

(m̂t

Pt

)]dt

]I The consumer’s budget constraint is:

E

[∫ ∞0

Qt

[∫pCt(p)φt(dp, dv) +Rtm̂t −Wtlt −Πt

]dt

]≤ m0

Πt: Profit income from holding of a fully diversified portfolioof claims on the individual firms plus any lump-sum transfersRt is the nominal interest rate and Rtm̂t represents theopportunity cost of holding cash





F.O.Cs

Money Holdings: e−ρt1

mt= λQtRt

Labor-Consumption: e−ρtc−γt c1/εt Ct(p)

−1/ε = λQtp

Labor Decision: e−ρtα = λQtwt

It can be shown that there is an equilibrium in which:

Rt = R = ρ+ µ

wt = αRmt

Thus log(wt) follows the same path as log(mt). Derivation of thisequation depends on crucial assumptions about utility function





Firms’ Decision

If the firm leaves its price unchanged, its current profit level is:

Ct(p)

(p− wt

vt

)If it chooses any price q 6= p its current profit level is:

Ct(q)

(q − wt

vt

)− kwt

Where the parameter k is the hours of labor needed to change theprice, the real menu cost





Firms’ Decision

Let ϕ(p, v, w, φt) denote the present value of a firm. This firmchooses a shock-contingent repricing time T ≥ 0 and ashock-contingent price q to be chosen at t+ T so as to solve:

ϕ(p, v, w, φt) = maxtEt

[∫ t+T

tQsCs(p)

(p− ws

vs

)ds

+QT .maxq

[ϕ(q, vt+T , wt+T , φt+T )− kwt+T ]

] (1)





Firms’ Decision

Using households’ F.O.Cs it is easy to show that:

Ct(p) = c1−εγt

(αp

wt

)−ε(2)

Qt+s = e−ρswtwt+s

(3)

Using (2), (3), we can express the Bellman equation (1) as:

ϕ(p, v, w, φt) = maxtEt

[∫ t+T

te−ρ(s−t)c1−εγt

(αp

ws

)−ε(p− ws

vs

)ds

+ e−ρTw

wT.max

q[ϕ(q, vt+T , wt+T , φt+T )− kwt+T ]

]





I We treat the special case in which the variance σ2m of themoney growth and wage process is zero, so that the driftparameter µ is simply the constant rate of wage inflation

I In this situation, there is an invariant distribution φ̃ for realprices xt = p

wtand idiosyncratic shocks v, thus the

consumption aggregate is:

ct =

[α1−ε

∫x1−εφ̃t(dx, dv)

]1/[γ(ε−1)]= c̄





Then the Bellman equation can be written as:

1

wϕ(wx, v, w) = max

tE

[∫ t

0e−ρsc̄1−εγt (αxs)

−ε(xs −

1

vs

)ds

+ e−ρT1

wT.′

maxx

[ϕ(wTx′, vT , wT )− kwT ]

]

Finally, the solution to the above equation is in the form ofφ(p, v, w) = wψ(x, v) and can be studied with familiar methods





I The two bounds in the Figure 1 determine the ”region ofinaction” on which the firm’s relative price x = p/w declinesat the rate µ because of deterministic wage growth, and itsproductivity level v moves stochastically

I When the boundaries are reached the price is changed to thedotted line on the figure

I Note that getting prices ”right” is more important whenproductivity shocks and hence quantities sold are high

I Next figure shows the necessity of including idiosyncraticshocks to describe the fraction of prices changed each month





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





Impulse responses are much more transient than a standard timedependent model would predict




Menu Cost vs Calvo Pricing

Comparing before and after distributions of individual prices toillustrate the reason for these different responses


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