Sticky Information The Impact of Different Information Updating Assumptions

BILL DUPORTAKAYUKI TSURUGASticky Information:The Impact of Different Information Updating AssumptionsVector auto-regressions applied to U.S.datafind that inflation and outputexperience persistent and hump-shaped responses to unanticipated monetaryshocks.One monetary model that can generate these responses employssticky information.Under sticky information,afirm may costlessly changeits nominal price every period;however,thefirm can only occasionallyupdate the information upon which it bases decisions.We show that thisresult hinges critically on the assumption that eachfirm receives informationupdates randomly.If,on the other hand,eachfirm updates its information atinfrequent butfixed intervals:(a)the output response is not persistent andonly weakly hump-shaped under one-year price contracts;(b)the inflationresponse suffers diminished persistence and a reduced hump-shape;(c)theinflation response remains unrealistic under longer durations.JEL codes:E0,E3Keywords:inflation,sticky information.A standard explanation for the real effects of changes in the money supply is nominal price rigidity.Iffirms face sufficiently large costs of nominal price adjustment,they will hold nominal prices constant for a period of time. An increase in the money supply will reduce the real price of these goods in the short-run,leading to an increase in sales and therefore output.A number of researchers seek to supplant this‘sticky price’explanation for non-neutrality with‘sticky information’on the part offirms.Under sticky information, price setters have entirelyflexible prices;however,they only infrequently update their price plans.11.Examples of this research include:Mankiw and Reis(2002)and Koenig(1996),which provide equilibrium models of sticky information;Carroll(2003),which uses survey data to study the frequency of updating information by households;and Ball,Mankiw,and Reis(2003),which studies optimal monetary policy under sticky information.We are grateful to Paul Evans,Huston McCulloch,Ricardo Reis,and Kenneth West for comments.Bill Dupor is an Associate Professor of Economics in the Department of Economics,the Ohio State University(E-mail:dupor.1Ȱ).Takayuki Tsuruga is a Graduate Stu-dent in the Department of Economics,the Ohio State University(E-mail:tsuruga.1Ȱ). Received July7,2004;and accepted in revised form March2,2005.Journal of Money,Credit,and Banking,Vol.37,No.6(December2005)Copyright2005by The Ohio State University1144:MONEY,CREDIT,AND BANKINGA compelling case for the sticky information approach is made by Mankiw and Reis(2002[MR]).They state that existing sticky price models are empirically counterfactual:the models cannot generate hump-shaped responses of inflation and output to a monetary shock.MR develop a model where sticky prices are replaced with sticky information.Under sticky information,afirm may costlessly change its nominal price every period;however,thefirm can only occasionally update the information upon which it bases decisions.MR demonstrate that a model with this information restriction is more successful at matching the stylized facts than one with sticky prices.This note shows that their result hinges critically on the assumption that each firm has a random iid probability of receiving information updates each period.If, on the other hand,eachfirm updates its information at infrequent butfixed intervals: (a)the inflation response suffers diminished persistence and an unrealistic hump shape;(b)the output response is not persistent and weakly hump-shaped under short fixed duration updating.Forfixed but longer duration price plans,the persistence and hump-shaped response of output are maintained.In addition,the main justification for using random information updating—computational simplicity—vanishes in the Mankiw–Reis sticky information formulation.The intuition for the change in dynamics can be explained with an example involving cars on a highway.Suppose highway drivers have two objectives:obeying the legal speed limit and holding their speed close to the average current speed on the road.Drivers,however,cannot immediately change their speed paths in response to new information regarding the average speed and the legal limit.We compare two information updating schemes denoted:the random duration scheme,which we describe later,and thefixed duration scheme.Under thefixed duration scheme, 10%of drivers receive the new information once every minute.This occurs in a staggered manner,i.e.one-tenth update in thefirst minute,the next tenth update in the second minute,etc.Consider a roadway in a steady state where all drivers move at the legal limit. Next,let the legal limit suddenly and surprisingly drop by ten miles per hour.Under thefixed duration,thefirst group of drivers learns not only that the speed limit fell, but also that half of all drivers will have this information infive minutes,all drivers will be informed in ten minutes,etc.Thisfirst group chooses quickly declining speed paths and future groups behave similarly.The average speed falls rapidly, reaching the new legal limit after ten minutes.Next,we describe the random duration scheme:every minute,10%of the drivers are informed randomly(with resampling).Consider the same fall in the legal limit as above.In this case,the decline in the average road speed is slower.Although thefirst group knows the new speed limit,this group rationally believes that a fraction will be driving at the higher previous speed even after ten minutes(due to resampling).These drivers also know that,even after20minutes,12.2%(ϭ(1Ϫ0.1)20)of drivers will be still moving at the initial higher speed.In response to the resampling,the group reduces their speed by less than the corresponding drivers under thefixed duration scheme.This keeps the average speed from declining.BILL DUPOR AND TAKAYUKI TSURUGA:1145 This effect is reinforced by the next group.Because thefirst group tempered their speed reduction,the second wave of drivers goes easier on the brake as well.This effect continues to be amplified as new drivers become informed.Overall,the speed falls more slowly under the random duration scheme than thefixed duration scheme.1.TWO INFORMATION UPDATING SCHEMES1.1Random Duration UpdatingSupposefirm i is the producer of good C t(i)under monopolistic competition, with the following demand curve:C t(i)ϭ[P t(i)րP t]ϪθC t,whereθϾ1.Thefirm chooses a price sequence at time t conditional on that period’s information set. Following the initial period,there is a constant probabilityλthat thefirm has the opportunity to reset its own price path.Thefirm chooses the sequence {P tϩj(i)}∞jϭ0to maximize:͚∞(β(1Ϫλ))j E t[u′(C tϩj)P tϩj{P tϩj(i)ϪW tϩj(i)}C tϩj(i)],jϭ0subject to its demand function.Here,we assume a‘yeoman farmer model’of production.At an interior optimum,P tϩj(i)ϭ(θθϪ1)E t(u′(C tϩj)C tϩj(i)W tϩj(i)րP tϩj)for jϭ0,1,2, (1)E t(u′(C tϩj)C tϩj(i)րP tϩj)For concreteness,let a household’s preference be u(C)–υ(L(i))ϭC1–σ/(1–σ)–L(i)1ϩψ/(1ϩψ)and W(i)/PϭCσ(P(i)/P)–θψCψ,whereσϾ0andψϾ0.Log linearizing both sides of Equation(1),we have:p tϩj(i)ϭE t[p tϩjϩαc tϩj]for jϭ0,1,2,…,(2) whereαϭ(σϩψ)/(1ϩθψ).Here,a lower case variable denotes the percentage deviation from its steady-state value.Next,we use the price index to relate individualfirm prices to aggregate inflation. The price level is P tϭ[͐10P t(i)1Ϫθdi]1ր(1Ϫθ).Since eachfirm’s probability of receiving the opportunity to update a price plan is independently and identically distributed,the overall price level is a sum of prices for the current period that have been set in the past.P tϭ[λ͚∞jϭ0(1Ϫλ)j(P t,j)1Ϫθ]1ր(1Ϫθ).Here P t,j denotes the time t price chosen by afirm that selected a price plan at t–j. Log linearizing this expression and using Equation(2),we have:1146:MONEY,CREDIT,AND BANKINGp tϭλ͚∞jϭ0(1Ϫλ)j E tϪj(p tϩαc t).Thefirst difference of this expression returns a law of motion for inflation:πtϭαλ1Ϫλc tϩλ͚∞jϭ0(1Ϫλ)j E tϪ1Ϫj[πtϩα(c tϪc tϪ1)].(3)Current inflation depends upon conditional forecasts,made in past periods,of current period inflation and consumption growth.Not surprisingly,sticky information can generate persistent and hump-shaped responses to an unanticipated monetary shock. Mostfirms do not reevaluate their plans in the initial periods following the shock.Thesefirms cannot adjust immediately to the newflexible price level.Moreover, afirm that can reset its plans in response to the shock will not move to theflexible price level,since its price path depends upon the non-updatingfirms’price.1.2Fixed Duration UpdatingNext,consider staggered inattentiveness with afixed duration.Suppose that a fraction1/N of allfirms chooses price plans every N periods.A price plan selected at period t byfirm i satisfies the followingfirst-order condition at an interior: P tϩj(i)ϭ(θ)E t(u′(C tϩj)C tϩj(i)W tϩj(i)րP tϩj)E t tϩj tϩj tϩjfor jϭ0,1,2,…,NϪ1.(4)Apart from the period j index of when the price is in effect,thefirst-order conditions under this scheme are identical to that of MR.The aggregate price index isP tϭ1N[͚NϪ1jϭ0(P t,j)1Ϫθ]11Ϫθ.Using Equation(4)and the definition of P t,we have:p tϭ1N͚NϪ1jϭ0E tϪj(p tϩαc t).(5)Next,we model the demand side of the economy by an exogenous stochastic process of nominal GDP.The growth rate of nominal GDP,∆m t,is∆m tϭρ∆m tϪ1ϩεt,(6) whereεt is iid over time.One can also interpret the variable∆m t as a measure of the growth rate of money supply under a cash-in-advance constraint,M t/P tϭC t.BILL DUPOR AND TAKAYUKI TSURUGA:1147 2.CALIBRATIONSWe next display the impulse responses of output and inflation to a particular monetary shock in the two models.Following MR,we selectρϭ0.5,αϭ0.1,λϭ0.25for simulations.Our choice ofλimplies that the expected duration of a newly set price for an individualfirm and the cross-sectional age of a stale price are four periods(1/λ). However,under thefixed duration model,they are different.The expected duration of a newly set price is N,while the cross-sectional age of a stale price is(1ϩ2ϩ…ϩN)/Nϭ(Nϩ1)/2.For comparisons,therefore,N can be determined under two criteria,based on the calibrated value ofλ.Thefirst criterion matches the expected duration of a newly set price for an individualfirm.2Our choice ofλimplies Nϭ4,because Nϭ1/λ.The second criterion matches the cross-sectional age of a stale price.In this criterion,(Nϩ1)/2ϭ1/λimplies Nϭ7.Both criteria are reasonable and we discuss them in turn.2.1Calibration I:Matching the Expected Duration of a Newly Set PriceFigure1plots the impulse responses of inflation and output to a one standard deviationεt shock3for both information updating schemes.4The sticky information model by MR exhibits significant output and inflation persistence.The peak re-sponses for inflation and output are periods seven and three,respectively.Under thefixed duration model,inflation sees dramatically reduced persistence and the peak response occurs one-year earlier.Output exhibits little persistence and peaks at period one.Inflation follows the path of the money growth rate from period four,because allfirms are informed about the change in money growth at period three.Moreover,output has returned to the steady state by period three. By this time,eachfirm is fully informed about the monetary disturbance and selects a price path that undoes the monetary non-neutrality.Thisfixed duration model shares the same difficulty of Fischer’s(1977)predetermined price model:the inability to generate real effects beyond the price friction’s duration.Figure2is consistent with our roadway example in the introduction.The nominal money supply unex-pectedly and gradually goes up,converging to the new steady state.Then,the optimal price for random durationfirms increases only moderately,because they know that other random durationfirms keep their optimal price low even after a year—the average period of duration.As a result,the price level(the average speed) increases more slowly than the path of nominal money supply(the legal limit)under the random updating scheme.2.This criterion is used as the standard assumption throughout the line of research on the sticky prices,as Kiley(2002)points out.3.The standard deviation ofεt is set to0.007,following MR.4.The details of the solution to thefixed duration model are available on request.1148:MONEY,CREDIT,AND BANKINGFig.1.Impulse Response Functions of the Output Gap and Inflation to a Positive One Standard Deviation(0.007) Shock–Matching the Expected Duration(Nϭ4)(Note:Both models assume sticky information.)On the other hand,the price level under thefixed duration scheme increases faster than that under the random duration scheme.The path of the price level catches up with the path of nominal money supply at exactly period three because allfixed durationfirms then know the new path of the nominal money supply.Given the fact that allfirms are informed at period three,allfixed durationfirms set their price much higher than random durationfirms,which is why inflation under the fixed duration scheme are larger than one under the random duration scheme. 2.2Calibration II:Matching the Cross-Sectional Expected Age of a Stale Price The next criterion matches the cross-sectional expected age of a price,instead of the expected duration of a newly set price.Figure3presents impulse responses of the output gap and inflation,i.e.Nϭ7.For the output gap,a hump shape can be recovered by increasing duration. When Nϭ7,the output gap peaks at period three and returns to the steady state at period six.Thus,it is possible to have a significant hump shape in output by changing the duration.Although a hump shape for output can be attained under our second criterion, the inflation response remains unrealistic.When Nϭ7,inflation gradually increases up to period six but decreases quickly in the next period.Why?Inflation must catch up with the money growth rate in period seven.Under an AR(1)money growth rate withBILL DUPOR AND TAKAYUKI TSURUGA:1149Fig.2.Impulse Response Functions of Price Level to a Positive One Standard Deviation(0.007)Shock(Nϭ4)ρϭ0.5,the money growth rate at period seven is almost zero.Therefore,inflation must decrease very quickly once allfirms are informed about the shock.2.3An Anticipated DisinflationThe change of modeling from the random to thefixed duration scheme affects MR’s prediction on anticipated and credible disinflationary policies.Suppose that the central bank announces a decrease in money growth two years in advance.They state that the sticky information model predicts output response consistent with conventional wisdom about disinflation:this disinflationary policy always causes a recession in the sticky information model.Under thefixed duration scheme,however,this disinflationary policy produces neither a boom nor a recession.This is because all price setters will have adjusted their plans when the policy begins.Thefixed duration model can produce a recession only when a disinflationary policy is announced quarters in advance beyond the length of rigidity(Ball1994).Even if the policy causes a recession,such a negative response of the output gap is much short-lived under thefixed duration scheme than under the random duration scheme.Therefore,we conclude that the cost of disinflation is zero or much smaller in the sticky information model under thefixed duration scheme.1150:MONEY,CREDIT,AND BANKINGFig.3.Impulse Response Functions of the Output Gap and Inflation to a Positive One Standard Deviation(0.007) Shock–Matching the Cross-Sectional Expected Age of a Stale Price(Nϭ7)3.THE MOTIV ATION FOR THE RANDOM DURATION SCHEMEEquation(3)has infinite lags on its right-hand side.These appear because each cohort’s price decision is necessary to compute the current inflation rate.On the other hand,the random duration scheme in a sticky price inflation equation does not require each cohort’s price decision.πtϭβE t(πtϩ1)ϩκc t,(7) whereβandκare parameters satisfying0ϽβϽ1andκϾ0.A key advantage of random duration overfixed duration in sticky price models is that the former does not require one to keep track of the distribution of price changes.The assump-tion that afirm is randomly‘tapped on the shoulder’and allowed to update its decision simplifies sticky price models because price distributions are not state variables;however,under sticky information,the random duration scheme does not reduce the model’s state space.Thus,an infinite discounted sum appears in Equation (3).Under thefixed duration scheme,the number of state variables equals N–1 under both sticky information and price models.The fact that the random duration scheme by MR does not simplify computation suggests that the decision between which timing to use mainly depends on which timing is empirically plausible for the cross-sectional distribution of information. Ourfinding is that the decision between the two is not innocuous.BILL DUPOR AND TAKAYUKI TSURUGA:1151 4.FINAL REMARKSThis note compares two models with slow diffusion of information.One is the random duration information updating scheme proposed by Mankiw and Reis(2002). The other is thefixed duration information updating scheme.Our comparison of the two information updating schemes depends on which of the two ways we calibrate the models for the comparison.When the two schemes are calibrated using the expected duration of a newly set price,thefixed duration model generates neither persistence nor a realistic hump shape in inflation and output dynamics.When the two models are calibrated using the cross-sectional expected age of a stale price,thefixed duration model performs better.It can generate a hump shape in output closer to the stylized facts.However,we emphasize that thefixed duration model is not successful because it overcomes the original criticism of early rigidity models,i.e.non-neutrality does not last beyond the length of the rigidity.Rather,it sets a long enough duration to match the slow response of inflation to shocks.A natural question arises:which scheme is more sensible for calibration?If the inattention towards pricing is well approximated by the random duration assumption (withλϭ0.25)for U.S.Fortune500companies,158of thesefirms would make pricing decisions using inflation and other macro data that is more than12months old.Given that business cycle peaks are difficult to forecast and that the average duration of post-WWII recessions is less than ten months,during a recession,many firms would be operating as if monetary policy was conducted in an expansion. Empirical plausibility of the cross-sectional distribution would be an interesting agenda in future work.5Moreover,sticky planning would likely have profound impact not only on mone-tary economics but also on optimal taxation,industrial organization and regulation, corporatefinance,and asset pricing.Therefore,finding evidence of widespread and long duration sticky plans by U.S.firms would be a next crucial step in this line of research.LITERATURE CITEDBall,Laurence(1994).“Credible Disinflation with Staggered Price-Setting.”American Economic Review84,282–289.Ball,Laurence,Gregory N.Mankiw,and Ricardo Reis(2003).“Monetary Policy for Inattentive Economies.”NBER Working Paper No.9491.Carroll,Christopher D.(2003).“Macroeconomic Expectations of Households and Professional Forecasters.”Quarterly Journal of Economics118,269–298.Fischer,Stanley(1977).“Long-Term Contracts,Rational Expectations,and the Optimal Money Supply Rule.”Journal of Political Economy85,191–205.5.Recently,Reis(2004)developed the inattentiveness model providing a micro-foundation for the random duration model.1152:MONEY,CREDIT,AND BANKINGKiley,Michael T.(2002).“Partial Adjustment and Staggered Price Setting.”Journal of Money, Credit and Banking34,283–298.Koenig,Evan F.(1996).“Aggregate Price Adjustment:The Fischerian Alternative.”Unpublished paper.Mankiw,N.Gregory,and Ricardo Reis(2002).“Sticky Information Versus Sticky Prices:A Proposal to Replace the New Keynesian Philips Curve.”Quarterly Journal of Economics 117,1295–1328.Reis,Ricardo(2004).“Inattentive Producers.”Unpublished paper.。