Economic Theory; Macroeconomics Sequence
Economic Theory; Macroeconomics Sequence ECON 714
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This 8 page Class Notes was uploaded by April Jerde on Thursday September 17, 2015. The Class Notes belongs to ECON 714 at University of Wisconsin - Madison taught by Kenneth West in Fall. Since its upload, it has received 53 views. For similar materials see /class/205139/econ-714-university-of-wisconsin-madison in Economcs at University of Wisconsin - Madison.
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Date Created: 09/17/15
Econ7l4 Spring 2009 K West Notes on Algebra to Solution of Monetary Model with Two Period Price Stickiness Half the intermediate goods rms set prices at the beginning of odd periods ie tis odd and the indeX i on the rm is between 0 and 12 the other half set prices at the beginning of even periods All intermediate goods rms hold their prices xed for two periods LetX denote the intermediate goods price set at the beginning ofperiod t to be reset at the beginning of period 2 We have 1 13 IPtquotdz39 39quot l X 39quotdi t l QOQSYWilll39w t Odd I 3Xiidz I 3X139quotdi 39quot r even For t odd or even 13 IPtquotdz39 39quot 2X139l VzXl39Wm39 2 1 21139quotXx1P139quot XPx139quot l39quot Upon taking logs this can be rewritten as 2 0 constant 1111111 CXP13911X139I7 6XP111xrl7z In the steady state X1P XP 1 Upon linearizing around the steady state and ignoring constants here and in the rest of the handout we obtain 3 17 V209 t gt94 The intermediate goods rm chooses X to maximize Et1 XzYn 39 WtzvitXtYit1 39 11Nn1 z1 2603109er WtAxxpJQUYJ XP1Xz Y 1 39 W1Ax1P1Xx Y 11 The rst order condition implies 4 X1 777391 Ez1WAzPle Wt1At1Pi1Yt1 Et1Pth Pi1Yt1 XIPt EIlWtAfPIYt WtlAt1PtlPtlPt1rYtl EXIIIPtlPITIYI1 Loglinearization then yields 5 x1 3917 05Et1Wt39at39pt l WHlaHlpHI ptl 3917 From previous discussion we know 71ny W 17 As well up to a log linearization n y ar Thus 6 IOFat Wt 3917 39 at39 Loglinearization to yield nya We have A Ni PXquotY for half the rms AN PX1quotY for the other half AW 5PXquot5PIQ1quotY e armyt1n 5 Xpn1vrxl 59XP11I7139xt1l in light of 3 a rst order linearization of the rhs yields a constant Using 6 and 3 5 becomes x1 O39SEtlDyt ytl 39 7atat1 l 0395xtlxt l 0395xtxtl 7 x1 05xz 1 05Et1xt1 05YEz 1Vzyz1 39 057Et1atat1 Digression Note that this can be rewritten as 7 xt39xtl Et1xt139xt Ex 1Vz39az yz139az1 Since yfa in the exible price model one can think of yta as an output gap 7 is thus qualitatively in the form of a Phillips curve in ation eXpected in ation ygtltmeasure of output gap Behavior in the household sector continues to yield the aggregate demand equation 8 y m 17 Equations 3 7 and 8 are three equations in the ve variables 17 x y m and a Once equations are speci ed for m and an we will have as many equations as variables K West Spring 2009 Econ 714 Loglinearization of the Stochastic Growth Model Recall that the rst order conditions and resource constraint may be written N r Yr 1 PW e1ltI Cr 1 Ct Ytl 2 1 BEA Cr 1 19 Km 15gt K t1 K Ct 3 K 15f17 All notation is as in the handout Stochastic Growth Model Let us loglineariEe equation by equation For a smooth function f we will repeatedly approximate fx around fx using fx z gg 05096 constant The constan will change from approximation to approximation and from equation to equation but will always be referenced simply as constant Lower case letters will initially denote logs of corresponding uppercase variables but will shortly be rede ned to indicate the deviation from steady state of such variables Equation 1 taking logs of both sides gives 4 1110p quott 39 Ylnll39eXPMJ 1119139ltP tyre Let fx lnlexpx and approximate fnt around f n where n lnA7 A7 the nonstochastic steady state value of M Innexpo a Innmm exp1exp391n 5 InaAB limb1mg 5 17 7l7391nt a constant yt ct Equation 2 Let us assume further that CMCt and 16YtKt 18 are lognormal and conditionally homoskedastic Then the right hand side of 2 is also lognormal and conditionally homoskedastic Recall that for a lognorrnal random variable x lnEx Elnx 5varlnx 2 is 6 Eachct constant E1n1eexpyn m 18 Letfx lnl6expxl8 and approximate y l k l around flnK7 Y where K Es the nonstochastic steady state value of KY Then 6 becomes 7 Eranct e constant leKE39111e1lt7 539118Etym m Equation 3 Rewrite this as 8 1neXpAkr15 y kt 1n1expc y Approximate the left hand side around the steady state value of KMKt E G E 1g the second term on the right hand side around the steady state value of CtYt EC Y This gives 9 1gg5Akzr e constant yr kt C Y1C Ycryt For future reference observe that C ElC Y rrge5g1e5g Equations 5 7 9 together with the production function yr at 16kt 6n and an exogenous equation for at say at constant pat1st are 5 dynamic stochastic equations in the ve variables yt ct nt k 1 and a To solve the system use the production function identity to eliminate yt from 5 7 and 9 For notational ease drop all constants reinterpreting all variables as deviations cs 77 from means For example n means lnM lnNquot Also replace with Then we get 10 nt v1atl0ktct v1 E lABl0l7y7 l l EtAc 1 sztaM0km0nm v2 lexK7 539111e1ltE391161 rtgtrgtsyutrtgtrg 12 Me t Ma ent k1 lrgrgPegy ta g81g1 Jo1455 stgtrtrgyKlexltgn Ag g81g5511 rrgeg81e1tgn Observe that with 81 100 percent depreciation equations 1012 yield the exact solution the 2 solution to these equations is r150 k 1ctatl6kt Let at follow an AR1 possibly with a unit root Etampat lplgl To solve the modelithat is nd time series processes for 11 0 and kt1 that are consistent with the rst order conditions 10127we shall use the method of undetermined coef cients In this technique for solving linear rational expectations models one guesses the form of the solution and then uses the rst order conditions to solve for the coef cients in the solution The form of the solution with 100 percent depreciation suggests that nt ct and k 1 will be eXpressed as functions of at and kt Inspection of 1012 indicates that the equations indeed are consistent with say 13 n Mk Man 14 ct m my 15 km kick Imam 16 am P l SHI39 To solve for the as use 10 to eliminate nt and 111 from 11 and 12 and use 12 and 16 to eliminate Eta 1 and Etk 1 from 11 Write the result as 17 Etcm aleJV dead do 18 km dkkkt dkaat dkcct39 The d coef cients may be computed in straightforward though tedious fashion from the coef cients in 10 l l 12 and 16 For example dkk q qevl e db M16v1 and d Agvlhze See Campbell Inspecting the Mechanism JME June 1994 for the formulas for the coef cients in 17 Note for future reference that dkkgtl since klgtl and A16l6v1gt0 dkclt0 since Aglt0 and v1q6 gt0 NeXt use 14 in 17 and 18 getting 19 Etc dckkt dead doomck Mat 20 km dkkkt dkaat dchcckt oaat Lead 14 by one period take eXpectations and use 20 and 16 to substitute out for Etk 1 km and Eram 21 Etcm Trad 116 TradWW ckdkcnckktnmat Empat Since 19 and 21 are both expressions for Etcm the right hand sides of the two must be the same This implies that the coef cients on kt in the two equations must be the same as must the coef cients on at For the coef cients on k this implies 22 no2dkc Wokdkk39dcc 39 dck 0 Campbell shows for some relevant parameter values one of the roots of this quadratic is real and negative the other real and positive Stability of the solution roots on or outside the unit circle requires choice of the positive root see below Since the d coef cients are known functions of the model parameters we can compute 75620 as a function of the model parameters Given 756k we may similarly compute 75 We can then use the results to substitute out for ct in 10 and 18 yielding nk 7cm 7 and 7th In the end we have eXpressed the TE s as functions sometimes quite messy of the model s parameters The stability condition referenced above Let X 1 amk 1 with XMHXtsM0 the solution The stability condition is lIHzl 0 lzl2l Since His lower triangular the two z s that solve this equation are the reciprocals of the diagonal elements of H From 18 and 14 H22 27 dkkdkc7tck from 16 Hll p By assumption llpl2l Since dkkgtl and dkclt0 75050 would imply lH22 lt 1 Hence a necessary condition for stability is 75620 Econ7l4 Spring 2009 K West The Calvo Model of Price Adjustment The Calvo model of price adjustment assumes that opportunities to adjust nominal prices arrive randomly and exogenously In particular each rm has a constant perperiod probability l of adjusting its price Thus the probability that a rm s period tprice will still prevail in period tl is i in period t2 is 2 and so on The eXpected time that a rm s price remains unchanged is ll In any given period a fraction l of rms adjust their prices Let 13 denote the price chosen by those rms The aggregate price level obeys 1 13 IPquotdi 139quot 1 131quot IPquotdi 39quot 1 131quot PHrU39quot A loglinearization of 1 around a zero in ation steady state yields apart from constants 2 171 17 lm We wish to solve for a relationship between 17 and other variables Clearly we need to model determination of 17 We take the shortcut of considering a partial equilibrium formulation noting at the end the connection to the aggregate model The eXposition follows Walsh pp225227 Letp denote the log of the eXible price which is the price the rm would set if it were free to choose a price each period More on this below Assume that eXpected costs of actual rather than eXible prices is W 39 2 3 05E120BI7itj39l7 j The eXpected contribution to l of 17 is 2 2 2 2 2 4 05E Pn39l i B Wit39l iH 5 g Pit39l in l The rst order condition with respect to pH is s pnzmsd imam 0 The solution 1 satis es 5 lB 27oB YEpfj 6 171B p7 swim Write the discrepancy between 17 and p as 2517337 for the moment taking 2 as an unmodeled variable Then 6 may be written 7 17 Home swim Now use equation 2 and equation 2 led one period to substitute out for 1 and E57 in equation 7 Upon de ning in ation 75217154 and rearranging we obtain 8 75 BEITEHI 9 21 7E1 1 B Recall that in the eXible price model we concluded that the rm sets its price to 17171WA So we have 2 E pi pt w a p where here as always we ignore constants Thus the variable 2 on the right hand side of the Phillips curve 8 is the log of real marginal cost or the the gap between the log real wage w p and the log of the marginal product of labor a If one maintains Calvo pricing but assumes that rms maximize the eXpected present discounted value of pro ts a log linearization still leads to 8 with 2 being real marginal cost See Walsh pp23423 8 Our assumptions on household behavior imply a direct mapping from real marginal cost to output In particular using yln y w p and the production function yna we have 2 yy a Since nally y a in the eXible price model we can write 2 yy eXible price output Let y denote eXible price output at and de ne the output gap as N F 9 y E yryx Then 8 may be written as 10 75BEM1 t Ki K E Y1 1 Bl
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