The dynamic new-keynesian model

Transcription

1 Chaper 4 The dynamic new-keynesian model A leas since Keynes, i has been hough ha in order o have real e ecs from moneary acions, i is key o have some degree of nominal rigidiy. The quesion we wan o explore is: can a model based on microfoundaions based on hese feaures describe some imporan feaures of he link beween moneary acions and he business cycle? Wha do we need in order o ge nominal rigidiies in he radiional, dynamic general equilibrium model? Well, we need some form of pricing power, for insance coming from monopolisic compeiion, and herefore some heerogeneiy among goods. The main acors of he DNK model are: agens=mks nal good inermediae labor pro money nom.bonds Household C W L F M M + T R B B = R nal rm j j dj = R inerm. rms j j dj W L F = Gov M M T = equilibrium = = = = = = households: make consumpion and labor supply decisions, demand money and bonds nal good rms: produce homogeneous nal goods from inermediae goods j inermediae good rm: use labor o produce inermediae goods j : Over each of his goods hey have monopoly power. Demand labor. Can se price of good j governmen: runs moneary policy. 4. Households There is a coninuum of in niely-lived individuals, whose oal is normalized o. They choose consumpion C ; labor L ; money M and nominal bonds B in order o maximize: 42

2 4.2. FINAL-GOODS FIRM 43 E X = C (L ) + ln M where E denoes he expecaion operaor condiional on ime informaion, is he discoun facor, subjec o: C + B = R B + W M M L + F + T where he wage is w W =, and bonds pay he predeermined nominal ineres rae R ; F denoes lump-sum dividends received from ownership of inermediae goods rms (whose problems are described below); he las hree erms indicae ne ransfers from he cenral bank ha are nanced by prining money. Here, is an appropriaely aggregaed price index for oupu and consumpion (ha we derive below) ha convers nominal ino real quaniies. Le = denoe he gross rae of in aion. Solving his problem yields rs order condiions for consumpion/saving, labour supply and money demand: R C = E + C (Euler) + w C C where m are real balances (M = ). = L = E + C + (LS) + (MD) m 4.2 Final-goods rm There is a nal-goods secor where a represenaive rm produces he nal good using inermediae goods j. Toal nal goods are given by he CES aggregaor of he di eren quaniies of inermediae goods produced: Z = j dj where >. The rm buys inpus j and produces he nal good in order o maximize pro s. Alernaively, he rm ries o minimize expendiure given he producion consrain. We can wrie he Lagrangean of he rm as: Z Z! L = j j dj + j dj

3 44 CHATER 4. THE DNAMIC NEW-KENESIAN MODEL Opimal choice of j = j = ; ha is j j = j dj j. = j = j This expression can be solved for he unknown value of he muliplier. Use he de niion of and use he soluion for j o wrie: Z j! dj A = Z j dj!. Because he producion funcion has consan reurns o scale, drops from boh sides of he expression, and we can hen solve for as: Z = j dj. represens he minimum cos of producing one uni of he nal-goods bundle (and, because of he consan-reurns-o-scale assumpion, is independen of he quaniy produced). For his reason we inerpre (he Lagrange muliplier) as he aggregae price index. The inuiion: ells us a he opimum wha is he value for he rm (in dollar erms) of relaxing he consrain on producion. 4.3 Inermediae goods The inermediae goods secor is made by a coninuum of monopolisically compeiive rms owned by consumers, indexed by j 2 (; ) The consrains Each rm, as we saw above, faces a downward sloping demand for is produc. I uses labor o produce oupu according o he following echnology: j = A L j Each producer chooses her own sale price j aking as given he demand curve. He can rese his price only when given he chance of doing so, which occurs wih probabiliy in every period. To summarize, each inermediae rm faces he following consrains:

4 4.3. INTERMEDIATE GOODS 45. The producion consrain: j = A L j j 2. The demand curve j = 3. The fac ha prices can be adjused only wih probabiliy. We follow Calvo (983) and assume ha every period only a random fracion of rms is opimally seing prices. Each period, his fracion is independen from he previous period. We can break his problem down ino wo sub-problems. As a cos minimizer and as a price seer roducer as a cos minimizer Consider he cos minimizaion problem rs, condiional on he oupu j produced. This problem involves minimizing W L j subjec o producing j = A L j (here is no sub-index on W since all secors where labor is employed mus pay same wage in equilibrium). In real erms his problem can be wrien as min L j W L j + Z ( j A L j ) [Z ] where Z is muliplier associaed wih he consrain. The rs order condiion implies W = Z A ; or, using j =L j = A : j = W (LD) L j Z noice ha his rs order condiion suggess han we can wrie he cos funcion in real erms as COST j W L j = Z j For his reason, we can hink of Z as real marginal cos; we can likewise de ne is inverse X = =Z as he markup. Given cos minimizaion, he rm akes Z as given, when choosing he oupu price, o which we urn now roducer as a price seer Digression: he problem wih exible prices (and he macro equilibrium wih exible prices) In order o warm yourself up, consider he problem of a monopolisic producer who has he chance o change her prices every period. The cos minimizaion problem is he same as before. On he revenue side, de ne r j j = he relaive price ha he producer charges. The maximizaion problem will be: max r j r j j Z j There is no subscrip on Z eiher because W is he only cos measure and is common o all rms.

5 46 CHATER 4. THE DNAMIC NEW-KENESIAN MODEL where j = r j. Opimal choice of r j will (r j j Z j ) =@r j = ; or: j j + r j + r @ Z j Z rj r j = j = Z = ha is, he relaive price would be a consan markup X cos. > over he real marginal Remark 6 This condiion is crucial because wih monopolisic compeiion bu exible prices we would derive a neuraliy resul similar o ha of Sidrauski model. In he symmeric equilibrium, j = ; hence Z = = < for all. Combining labor supply (equaion X LS) and labor demand (equaion LD) and imposing marke clearing = C would give w = L labor supply = A =X labor demand using L = =A in he symmeric equilibrium we will have: = A + X so ha oupu in he model is a funcion only of echnology (and money and nominal rigidiies do no have any e ec on he dynamics of he model around he seady sae, like in he Sidrauski model wih uiliy separable in consumpion and real balances). In saic erms, oupu would be subopimally low The problem wih sicky prices To begin wih, a any poin in ime if some inermediae good producers can change prices and ohers canno, he average price level will be a CES aggregae of all prices in he economy, and will be = + ( ) ( ) (*) where is previous price level, and is avg price level chosen by hose who have he chance o change prices (Call he rese price). I is a hese guys ha we look now. Consider he inermediae goods producer who has a chance o rese prices a ime. The demand curve is: j+k = j= +k +k for any period k for which he will keep ha price.

6 4.3. INTERMEDIATE GOODS 47 His maximizaion problem is: X max () k j E ;k j +k k= ;k = (C =C +k ) Z +k j+k (#) where Z is he real marginal cos. represens he probabiliy ha he price j chosen a will sill apply in laer periods. This expression is he expeced discouned sum of all pro s ha he price seer will make condiional on his choice of j and weighed by how likely j is o say in place in fuure periods. A ime ; he price seer chooses j o maximize pro. Di ereniae he pro funcion wih respec o j o obain X k= X () k E k= () k E ;k j+k +k ake j+k +k X k= X k= + +k j Z j ou, isolae elasiciy of j+k wr j j+k ;k + j +k j+k () k E ;k j+k +k () k E ;k Z +k +k j + Z +k +k j muliply everyhing by j j Z +k +k j j+k = j = In equilibrium, all he rms ha rese he price choose he same price (and face he same demand), hence j = These wo expressions ener he equilibrium (using X = is ha we derived above, he oher is: X k= () k E ;k +k +k XZ +k!# = = seady sae markup ). One = (**) Use Z+k n Z +k +k o rearrange he expression above o obain: X () k E ;k +k X = X () k E ;k +kz +k k= = X +k k= ()k E ;k k= +k Zn +k +k k= ()k E ;k +k +k = X X k= ;kz n +k

7 48 CHATER 4. THE DNAMIC NEW-KENESIAN MODEL where ;k = ()k E [ ;k +k +k] k= ()k E [ ;k +k +k]. This expression says ha he opimal price is a weighed average of curren and expeced fuure nominal marginal coss. Weighs depend on expeced demand in he fuure, and how quickly rm discouns pro s. Therefore you can noice he following: Under purely exible prices, = : he markup is a consan. = XZ n and opimal prices are a muliple X of he curren marginal cos. When > ; he opimal price depends on fuure expeced values of aggregae variables as well as fuure nominal marginal coss Z+k n. u di erenly, one can see ha all he ucuaions in he markup are due o rms being unable o adjus prices. 4.4 The equilibrium 4.4. Closing he model We need o combine everyhing, impose marke clearing, and linearize around he seady sae. Toal oupu in economy is: Z = j dj Z = (A L j ) dj I is no possible o simplify his expression since inpu usages across rms di ers. However he linear aggregaor = R jdj is approximaely equal o wihin a local region of he seady sae. Hence for local analysis we can simply use = A L Goods marke clearing is simply = C : Trivially, bond marke clearing implies B = Moneary policy We assume ha he cenral bank policy ses he dynamics of money supply in a way o achieve a arge level of he ineres rae. This way, he money demand equaion becomes redundan since i only serves o deermine he behavior of endogenous money. To beer gain insigh ino his, consider he money demand equaion of he model. Because uiliy is separable beween consumpion and money balances, his is he only insance where money appears. Given ha M is a funcion of R ; his implies ha he cenral bank can always choose he supply of money o implemen he ineres rae i desires. From M = C R R

8 4.4. THE EQUILIBRIUM 49 we can see ha, for given values C and, his equaion implies a one-o-one mapping beween M and R : We nd i convenien o work wih so called ineres rae rules, closing he model specifying an exogenous process for R raher han M : The equilibrium in levels Le us look a he equaion summarising he model: R = E = A Z (2) = + ( ) E X X ;kz +k +k (3) k= R = R r rr + Z Z () z! r r; (4). Eq. is he aggregae demand equaion: i combines goods marke clearing wih he Euler equaion for bonds 2. Eq. 2 is he equilibrium in he labor marke. Take labour demand (LD) and labor supply (LS) and impose marke clearing. Then equae (LD) and (LS) so as o eliminae of he real wage w from ha expression. Whenever you have L; remember o replace i wih =A. 3. Eq. 3 is he equaion ha describes how he aggregae price level is a weighed average of () previous price level and (2) rese prices, which are in urn a funcion of fuure expeced marginal coss. 4. The las expression is he moneary policy rule. rr is he seady sae real ineres rae. We assume ha he cenral bank chooses money supply so as o se he nominal ineres rae o be a funcion of previous ineres rae, curren in aion and curren real marginal coss (as a fracion of heir seady sae value). r is a smoohing erm. This is a Taylor rule, from John Taylor of Sanford Universiy, who was he rs o noice in a 993 seminal paper ha cenral banks se he ineres rae as a funcion of in aion and oupu gap (oupu gap=deviaion of oupu from is naural rae). The las erm r; represens a moneary policy shock. Noice ha under his forumulaion, he arge (seady sae) in aion rae is. This happens because in seady sae from equaion he real ineres rae is rr = =. Hence R = rr in seady sae (from ) where is he gross in aion rae. Hence 4 in seady sae holds only if =, which implies a consan price level.

9 5 CHATER 4. THE DNAMIC NEW-KENESIAN MODEL 4.5 The log-linear equilibrium 4.5. Linearizing he hillips curve Use +k = ( = +k ) +k and cancel ou in numeraor and denominaor o obain: = X k= ()k E ;k k= ()k E ;k +k +k +k +kz n +k To gain insigh ino his expression, i is convenien o loglinearize i. Inuiively, we can see ha numeraor and denominaor only di er up o a muliple given by Z+k n, which in urn muliplies () k ( ). Hence we can expec ha in log-linearising ; and will cancel ou and disappear. Use he seady sae resul ha Z n X =. Rearranging and dividing by : X k= () k E ;k +k +k = X X k= () k E ;k +k +k Zn +k LHS rs X b b () k + k= k= X () k E b;k + b +k + ( ) b +k k= RHS nex X b () k + X X () k k= k= Z n E b;k + Z cn +k + b +k + ( ) b +k = X b () k X + () k E b;k + Z cn +k + b +k + ( ) b +k k= where we have used XZn =. Hence, using c Z n +k = b +k + b Z +k b k= ()k = k= ()k E b+k + Z b +k b = ( ) k= ()k E b+k + Z b +k (@) Equaion (@) simply saes in log-linear erms ha he opimal price has o be equal o a weighed average of curren and fuure marginal coss, weighed by he probabiliy ha his price will hold in laer periods oo. So you assign weigh o oday, weigh o omorrow, 2 2 o he day afer omorrow, and so on. Noice he complee forwardlookingness of his expression, and he fac ha hese weighs need o be normalized (he sum of all of hem is ; whose inverse premuliplies he summaion - weighs sum up o one -)

10 4.5. THE LOG-LINEAR EQUILIBRIUM 5 Using b b = ( ) b ; he expression above can be rearranged as follows: b = ( ) E b + Z b + b+ + Z b (:::) + ::: b = ( ) b + Z b + E b + ( ) b = ( ) ( ) b + Z b + ( ) b + b b = ( ) ( ) b + Z b + E + b b b b = ( ) b + ( ) ( ) b + Z b + E + b b b b = ( ) b b ( ) b + ( ) ( ) Z b + E + b b + ( ) b b = ( ) b + E b + + ( ) ( ) b Z ( ) ( ) b = E b + + bz This equaion is nohing else bu an expecaions augmened hillips curve, which saes ha in aion rises when he real marginal coss rise. I akes a while o derive, bu again i is nohing else bu an aggregae supply curve for he whole economy. @ Z b =@ = ( @ Z b =@ < he higher ; he higher he weigh o fuure b Z s, and he lower oday s elasiciy o curren marginal cos he higher ; he higher he chance ha I will be suck wih my price for a long period, and he higher he elasiciy of b o Z b. However, few prices will be changed in he aggregae, herefore aggregae in aion will no be sensiive o he marginal cos The remaining equaions Equaions (), (2) and (4) are already linear in logs. We assume ha a and e follows AR () processes The complee log-linear model From now on, we denoe wih lowercase variables deviaions of variables from heir respecive seady saes. I now work in erms of he markup raher han he real marginal cos. When we log-linearize he 4 expressions above, wha we obain he following sysem:

11 52 CHATER 4. THE DNAMIC NEW-KENESIAN MODEL y = E y + (r E + ) y = + z + + a ( ) ( ) = E + + z + u r = r r + ( r ) (( + ) + z z ) + e Remark 7 The linearized equaions are in he Malab le, where we use x = z (he real marginal cos Z is he inverse of he markup X in levels, and x = z in logs - see equaion LD). dnwk:m and dnwk_go:m simulae his model. I has 7 equaions raher han four, bu you can forge abou hree of hem. One equaion is capial demand if you exend his model o have capial as well (you can se he weigh on K o be arbirarily small in he producion funcion, so ha equaion does no coun); he oher says ha = C; anoher de nes as he marginal uiliy of consumpion. I is someimes convenien o call y n a new variable ha de nes he equilibrium level of oupu (he naural oupu) ha would prevail under compleely exible prices ( = ). This way z can in fac be eliminaed. In fac, If rms were able o adjus prices opimally ( )( ) each period, z = (since ) ) and we would be able o de ne he exible price equilibrium values for real ineres rae rr (nominal minus expeced in aion) and oupu, which we call heir naural raes: y n = + a rr n = + (a E a + ) We can hen derive an expression for x as a funcion of he gap beween exible price and sicky price equilibrium, ha is: x = ( + ) (y n y ) Hence x is posiive whenever y is below y n ; oupu is below is naural level. I is for his reason ha we someimes refer o x as he oupu gap, since x is proporional o he shorfall of oupu from is naural level. y n is an exogenous variable, since i depends only on echnology. Wih his convenion, he dynamic new-keynesian model can be rewrien as: y = E y + (r E + ) (a) = (y y n ) + E + + u (b) r = r r + ( r ) (( + ) + x (y y n )) + e (c) ( )( ) where = ( + ) ; = =. Some auhors have also posulaed cos push shocks u, ha push in aion up. Some auhors refer o he sysem made by (), (2), (3) as he benchmark dynamic-new keynesian model.

12 4.6. THE DNAMIC EFFECTS OF TECHNOLOG AND MONETAR SHOCKS 53 TECNO SHOCK R X.5.5 π INFLATION SHOCK MONETAR SHOCK years years years years Figure 4.: Simulaions from dnwk.m, exible (riangles) versus sicky price (circles) model 4.6 The dynamic e ecs of echnology and moneary shocks The Malab programs in my webpage will allow you o analyze he dynamics of his model by means of impulse response funcions. The plo compares he responses ha obain under exible prices versus sicky prices. I was generaed wih dnwk:m 2 Technology shocks.. Following a rise in echnology, marginal coss fall. Since no all prices are free o fall immediaely, markups will rise. A fracion of ex price rms will lower heir prices and hire more facors of producion. A fracion of xed price rms will be unable o lower heir prices and o increase heir sales and herefore will hire less facors of producion. roducion rises less han wih exible prices 2 a =.5 ; e =. ; u =. ; =.99; =.5 ; =.75 and.; X =. ; = ; r =.8 ; = 2 ; x =. ;

13 54 CHATER 4. THE DNAMIC NEW-KENESIAN MODEL 2. The heory of endogenous markup variaions provides he crucial link ha allows he concerns of RBC models and convenional moneary models o be synheised. In addiion, he markup direcly measures he exen o which a condiion for e cien resource allocaion fails o hold. Moneary shocks. A moneary conracion leads o drop in oupu, rise in he nominal ineres rae, fall in in aion, and a rise in he oupu gap. These predicions, which are qualiaively in line wih he VAR evidence, are hard o obain in he exible price model. In aion ( cos-push ) shocks In aion shocks are imporan in his seup because hey generae a rade-o beween oupu gap versus in aion sabilizaion.