Monetary Theory ECN 235A
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This 27 page Class Notes was uploaded by Madie Schinner on Tuesday September 8, 2015. The Class Notes belongs to ECN 235A at University of California - Davis taught by Staff in Fall. Since its upload, it has received 44 views. For similar materials see /class/191901/ecn-235a-university-of-california-davis in Economcs at University of California - Davis.
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Date Created: 09/08/15
Introduction to State Dependent Pricing Model Takeshi Yagihashi ECN 235 Monetary Theory Novem ber 2007 El 5 3 RCquot Introduction FAQ on state dependent pricing model for those who cannot wait Q what is Statedependent pricing A here is the basic idea firms constanty reevauating the bene t of price adjustment against the current menu cost of a price Change narrow definition Q what is the relevant quotstatequot vector A entire firms39 history of pricing decision summarized as a fraction of firms who adjust do not adjust in each period Q what is empirical relevance of this model A so far mixed But many people are now working on this literature key papers 0 Caplin and Leahy 1991 O Dotsey King and Wolman 1999 we cover this 0 Golosov and Lucas 2007 Dotsey and King 2005 has a nice literature review model Q what is wrong with Time dependent pricing TDP typical TDP model a la Calvo 1983 Yun 1996 Pt 7 e ErEf ioP AMHMCrH P7 Yrj ie 1 co 39 PH 671 EtEjoP Ajt39 p1 Ytj 1 11 Pt Pz1 dz 2 0 El 5 3 RCquot Q what is wrong with Time dependent pricing TDP model Woodford 2003 TDP model is a good approximation to SDP model as long as the researcher39s interest in the present study is in the identification of better monetary policies Within the class of policies under which inflation is never very great p142 Klenow and Krytsov 2005 myjob market paper coming soon The model Dotsey King and Wolman quotStateDependent Pricing and the General Equilibrium Dynamics of Money and Output QJE 1999 o households 0 firm 0 government households households maximize their lifetime utility 1 170 max Er i0 D R 21 Wt N Dz Prom 62 7 iii 4 NW 3 st C FOC implies C717 Ctilf BRtrl1Et RNA r1 47 KN W Cg note DKW also has money demand firms firms maximize their lifetime value They incur random xed labor cost Wit when adjusting price VPrv Ct maXVzr V0 tht 4 where V2 ZPzr EtiAt1 VPzz1Ct1i P VOt m x AP ErAr1V pritv Ct1gt p r1 and ZPlyt P21 Mcri Yzt 5 xed adjustment cost C1 CgtanC3C 7 C4 which satis es CO 0 lt CX lt l CB forX E 0 B B lt oo 1 G xi n1 nuaz n nmn n ma u must H mm m Bun B EIEIEIEI u EIEIEIE u ume H mu m casts in unit nllzhnr xi Figure Comparison of the Distribution of the Fixed Cost Il firms pricing mechanism assign quotbinquot lt J I 31491 Mmmu Date H I 10min firms pricing mechanism endogenous change in fractions a probability of nonadjustment 11 17 txjt o for borderline firm in each bin Vz V0 7 Hpinns down the prob of adjustment nonadjustment a total fraction of firms who adjust their prices in current period t J Wu Elkfelt j1 firms pricing mechanism endogenous change in fractions cont 0 total fraction of firms who adjust their prices in current period t J Wu Elkfelt 11 0 fraction of firms who choose not to adjust prices within their originally assigned bin 11 ne3r o end of period fractions wj can be dynamically related to the beginning of period fraction 91 9r1 Wm firms for priceadjusting firm 0 A H V0 20 0 1r1 31 V0z1 Wr1d1z1 t A W1z1 I V1r1 where a 1 Ejyt Cz lz forjlJ71 0 FCC of pricing decision is 7 32P Ar1 aV1r1 0 3P 55 11z1 At 3 P firms for nonadjusting rms lJ 71 A H V1 Zjr j1r1 5 31 V0z1 Wt1dj1t1 t A j1z1 52 VjHIH fOFJ 0J i 2 A VJ71t ZJ71 3 31 Vor1 Wt1EJt1 for J 1 I envelope condition avjl 7 32042159 E L1 At Bj1z1 P z1ij for 7 0 iapfij apfij r L j1r1 At apfij J 3mm azltPLJ1v 5 for J 71 BPLJH BPLJH firms recursive substitution yields J71 wt P Pquot i 6 Et 0 lt AjtjMCti lt 3 Yti 6 I 7 7 e 71 7 P 1 Er 1401 35 AIM lt i5 Yr price level is given as 1 1 J71 Pr ng more 7 General Equilibrium other equations representing preference and technology nAmeM7WV J71 P 7 r N 7 ZenaZr 11 e MWM w for simpli cation DKW uses constant capital linear technology in labor and monetary aggregate rule compute steady state values The programming algorithm written by DKW does the following 0 solves a nonlinear system of equations analytically for J 1 pencil and paper method uses the result to generate a vector of starting guesses for the J 2 case fsolve nlsys The program then uses the J 2 solutions to generate J 3 starting guesses continues in this manner until a stopping criterion is triggered compute steady state values solve for J 4 equations with J 4 unknowns Use the expressions for J71 lee 0 price level P 2 Wm Pfij j0 J71 w t P e E 0 H30 ArmVIC Pt m l 71 J71 my ti PH 6 520 Am pt39 Y we 0 real marginal cost pf J71 0 amount of labor used in price adjustment Np E 9133th 11 compute steady state values solve for J 4 equations with J 4 unknowns Use the expressions for o wageoutput ratio 1 MEL MCtAtl Ntftvf Aerl l1N WV 0 borderline rm s xed cost function in each bin Jl equations ijt V0 thjyt 0 value for the priceadjusting rm V0 20 n1 A V0z1 Wr1ELz1gtl 1711 5231 V1144 compute steady state values what pinns down steady state values 0 preference and technology parameters pt 0 steady state inflation 0 xed cost parameters C1 C4 C1C2tanC3CC4 9 stopping criterion 0 cost function inverse function of the cdf B arctanbz 7 17139 arctand7 l 62 arctanb7d7 arctand7r the value of B b d is implicitly defined by C1 C4 0 the maximum unit cost of adjustment for individual firm occurs when 2 1 7gt 61 B so maximum total cost is WB a if at T period v0 7 VT WB then T is the maximum horizon J T o if v0 7 VT lt WB then you still keep increasing the T 9 cf in MATLAB options optimset MaxFunEvals 10 4 tolfun lOAS LargeScale oFF I l compute steady state values We obtain the 55 values of the maximum horizon a firm could fix their price J vector of fractions and transition probabilities 9147 o a 0 vector of transition probabilities 13411 0 optimal relative price p o example assume BurnsMiller period 7155 E W71 66 and VolckerGreenspan period 7155 34 All other parameters are the same across period Key Steady State Values BurnsMiller VolckerGreenspan Maximum number of bin J Fraction of nonadj firms 1750 0510 0584 1015 1010 Optimal relative price E note parameterization based on Cogley and Yagihashi 2007 D namics Ioglinearize the behavioral equations and express it into statespace form Vt 0 315 Eer 5 Tlstil hez er m iidN0 2g D namics impulse response to money supply shock DKW 1999 p674 summ E E E E E w 14 t 5amp5le n In I z a I inlamunu um D namics simulation based on couterfactual scenario and 2007 name amics selected moments 0 standard deviation of 7139 y name o autocorrelation of 7139
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