The dynamic new-keynesian model

Transcription

1 Chaper 4 The dynamic new-keynesian model Recen years have seen an explosion of models in which here are nominal rigidiies; hese models have nesed he RBC model as a special case. A leas since Keynes, i has been hough ha in order o have real effecs 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 Þnal good inermediae labor proþ money bonds Household Pc WL PF M M + PT RB B = Þnal Þrm PY R P jy j dj = R inerm. Þrms P jy j dj WL PF = Gov M M PT = equilibrium = = = = = = households: make consumpion and labor supply decisions, demand money and bonds Þnal good Þrms: produce Þnal goods Y from inermediae goods Y j inermediae good Þrm: use labor o produce inermediae goods Y j. Over each of his goods hey have monopoly power. Demand labor. Can se price of good Y j governmen: runs moneary policy. 4. Households There is a coninuum of inþniely-lived individuals, whose oal is normalized o. They choose consumpion c, labor L, money M and bonds B in order o maximize: 3

2 32 CHAPTER 4. THE DYNAMIC NEW-KEYNESIAN MODEL E X = µ c ρ β ρ η (L ) η + χ ln M P where E denoes he expecaion operaor condiional on ime informaion, β is he discoun facor, subjec o: c + B = R B + W L + F M M + T P P P P where he wage is w W /P, and bonds pay he predeermined nominal ineres rae R ; F denoes lump-sum dividends received from ownership of inermediae goods Þrms (whose problems are described below); he las hree erms indicae ne ransfers from he cenral bank ha are Þnanced by prining money. Le Π P /P denoe he gross rae of inßaion. Solving his problem yields Þrs order condiions for consumpion/saving, labour supply and money demand: µ R c ρ = βe Π + c ρ (Euler) + w c ρ where m are real balances (M /P ). c ρ = L η = E µ β Π + c ρ + (LS) + χ (MD) m 4.2 Final-goods Þrm There is a Þnal-goods secor where a represenaive Þrm produces he Þnal good Y using inermediae goods Y j. Toal Þnal goods are given by he CES aggregaor of he differen quaniies of inermediae goods produced: µz Y = Y j dj where >. The Þrm buys inpus Y j and produces he Þnal good in order o maximize proþs, aking P j as given: max Y Z P j Y j dj Y j P where P is an index (o be deermined) ha convers nominal expendiures ino real expendiures.

3 4.3. INTERMEDIATE GOODS 33 Opimal choice of Y j solves: " µz From Y j = Y j Y j dj P µz Y j dj {z } Y / Y j = µ Pj P Z Y # P j Y j dj = Y j = P j P Pj P Y (demand for each inpu) use CES o obain: Y j = P = µ Pj P µz and solving for P P j R Y j Y j dj Y j dj µz = Pj dj P represens he minimum cos of achieving one uni of he Þnal-goods bundle Y. For his reason we inerpre P as he aggregae price index. In equilibrium proþs in his secor will be equal o zero: his occurs because he producion funcion in he Þnal goods Þrm problem has consan reurns o scale, herefore from he Euler s heorem here canno be proþs. 4.3 Inermediae goods The inermediae goods secor is made by a coninuum of monopolisically compeiive Þrms owned by consumers, indexed by j (, ) The consrains Each Þrm, as we saw above, faces a downward sloping demand for is produc. I uses labor o produce oupu according o he following echnology: Y j = A L j Each producer chooses her own sale P j aking as given he demand curve. He can rese his price only when given he chance of doing so, which occurs wih probabiliy θ in every period. So, how many consrains do inermediae goods Þrms face?

4 34 CHAPTER 4. THE DYNAMIC NEW-KEYNESIAN MODEL. he producion consrain: Y j = A L j 2. he demand curve Y j = Pj P Y 3. he fac ha prices can be adjused only wih probabiliy θ. We follow Calvo (983) and assume ha every period only a random fracion of Þrms is seing prices. Each period,hisfracionisindependenfromhepreviousperiod. We can break his problem down ino wo sub-problems. As a cos minimizer and as a price seer Producer as a cos minimizer Consider he cos minimizaion problem Þrs, condiional on he oupu Y j produced. This problem involves minimizing W L j subjec o producing Y 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 P L j + Z (Y j A L j ) [Z ] where Z is muliplier associaed wih he consrain. The Þrs order condiion implies: Y j L j = Z W P X W P (LD) noice ha his Þrs order condiion suggess han we can wrie he real cos funcion as COST = W P L j = Z Y j For his reason, we can hink of Z as real marginal cos; we can likewise deþne is inverse X =/Z as he markup. Given cos minimizaion, he Þrm akes Z as given, when choosing he oupu price, o which we urn now Producer as a price seer Digression: he problem wih ßexible prices (and he macro equilibrium wih ßexible 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, deþne r j P j /P he relaive price ha he producer charges. The maximizaion problem will be: max r j Y j Z Y j r j

5 4.3. INTERMEDIATE GOODS 35 where Y j = r j Y. Opimal choice of r j will imply µ Yj Y j + r j + r j Y j Yj rj Yj Z rj rj Z Y j r j r j Y j = Yj = r j r j = P j P = Z ha is, he relaive price would be a consan markup X over he real marginal cos. Remark 9 This condiion is crucial because wih monopolisic compeiion bu ßexible prices we would derive a neuraliy resul similar o ha of Sidrauski model. In he symmeric equilibrium, P j = P, hence Z = = < for all. Combining labor supply (equaion X LS) and labor demand (equaion LD) and imposing marke clearing Y = C would give w = Y ρ L η labor supply = A /X labor demand using L = Y /A in he symmeric equilibrium we will have: µ A η ρ+η Y = X so ha oupu in he model is a funcion only of echnology. 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 P = θp +( θ)(p ) (*) where P is previous price level, and P is avg price level chosen by hose who have he chance o change prices. I is a hese guys ha we look now. Consider he inermediae goods producer who has a chance θ o rese prices a ime. CallP he rese price. The demand curve is: Y j+k = P j/p +k Y +k for any period k for which he will keep ha price. His maximizaion problem is: max Pj k= (θβ) k E µ Λ,k µ P j P +k Z +k Λ,k =(C /C +k ) ρ Yj+k (#)

6 36 CHAPTER 4. THE DYNAMIC NEW-KEYNESIAN MODEL where Z isherealmarginalcos. θ represens he probabiliy ha he price Pj chosen a will sill apply in laer periods. This expression is he expeced discouned sum of all proþs ha he price seer will make condiional on his choice of Pj and weighed by how likely Pj is o say in place in fuure periods. A ime, he price seer chooses P o maximize proþ. Differeniae # wih respec o Pj/P +k rj (he relaive price) o obain k= " (θβ) k E k= µ (θβ) k Y E Λ,k Yj+k + rj j+k Y j+k Z rj +k rj ake Yj+k ou, isolae elasiciy of Yj+k wr rj k= k= Λ,k Y j+k à + r j Y j+k (θβ) k E Λ,k Y j+k (θβ) k E Λ,k Y j+k Yj+k rj µ Z +k r j + Z +k r j muliply inside brackes by µ P j P +k r j rj Yj+k Z +k = Y j+k r j = =!# = In equilibrium, all he Þrmsharesehepricechoosehesameprice(andfacehesame demand), hence P j = P These wo expressions ener he equilibrium (using X = is ha we derived above, he oher is: (θβ) k E Λ,k Y+k k= = seady sae markup ). One µ P X Zn +k = (**) P +k P +k where Z +k = Z+k n /P +k. Rearrange he expression above o obain: P P k= = X (θβ)k E Λ,k Y+k P +k Zn +k P k= (θβ)k E Λ,k Y+k P = X X +k where φ,k = +k P +k] +k] (θβ)k E [Λ,k Y P k= (θβ) k E [Λ,k Y+k P k= φ,kz n +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 Þrm discouns proþs. Therefore you can noice he following: Under purely ßexible prices, θ =:he markup is a consan. P = XZ n and opimal prices are a muliple X of he marginal cos.

7 4.4. THE EQUILIBRIUM 37 When θ>, he opimal price depends on fuure expeced values of aggregae variables (P +k,y +k ) as well as fuure nominal marginal coss Z+k n. Pu differenly, one can see ha all he ßucuaions in he markup are due o Þrms 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 Z Y = Y j dj = (A L j ) dj I is no possible o simplify his expression since inpu usages across Þrms differs. However he linear aggregaor Y = R Y jdj is approximaely equal o Y wihin a local region of he seady sae. Hence for local analysis we can simply use Y = A L Goods marke clearing is simply Y = 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 money demand: In log-linear erms his becomes: M P = χc ρ R R br = ψ ρ bc bm (md) where ψ is some posiive coefficien. So far, we have speciþed cenral bank policy as conrol over a moneary aggregae, for insance, for consan money supply: cm = However, one can hink of several oher moneary rules: for insance, if he cenral banks wans o peg he ineres rae, i simply pegs nominal consumpion growh so ha: cm = bp + ρ bc (ms)

8 38 CHAPTER 4. THE DYNAMIC NEW-KEYNESIAN MODEL bu his implies ha we can rewrie money marke equilibrium as: br = and his is an ineres rae rule. The poin I wan o make is: if money eners separably he uiliy funcion, we can forge abou money demand in he model, and we can close he model by specifying any process for he policy insrumen (M or R) we like o consider The equilibrium in levels Le us look a he equaion summarising he model: Y ρ µ R P = βe P + Y ρ + Y ρ η = A η /Z (2) P = θp +( θ) E X X φ,kz +k P +k (3) k= Ã µ P ()! φr r, (4) R = R φ r P +φπ Z φ z. is he aggregae demand equaion: i combines goods marke clearing wih he Euler equaion for bonds 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 Y/A. 3. is he equaion ha describes how he aggregae price level is a weighed average of () previous price level P and (2) rese prices P, which are in urn a funcion of fuure expeced marginal coss. 4. The las expression is he moneary policy rule. 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 inßaion and curren real marginal coss. This is a Taylor rule, from John Taylor of Sanford Universiy, who was he Þrs o noice in a 993 seminal paper ha cenral banks se he ineres rae as a funcion of inßaion and oupu gap (oupu gap=deviaion of oupu from is naural rae). The las erm r, represens amonearypolicyshock.

9 4.5. THE LOG-LINEAR EQUILIBRIUM The log-linear equilibrium 4.5. Linearizing he Phillips curve Use Y+k =(P /P +k ) Y +k and cancel ou P in numeraor and denominaor o obain: P = X P k= (θβ)k E Λ,k P +k Y +k P+k Z +k P k= (θβ)k E Λ,k P +k Y +k To gain insigh ino his expression, i is convenien o loglinearize i. Inuiively, we can see ha numeraor and denominaor only differ up o a muliple given by P +k Z +k,which in urn muliplies (θβ) k ( θβ). Hence we can expec ha in log-linearising Y, P and Λ will cancel ou and disappear. Rearranging and dividing by P : P P k= (θβ) k E Λ,k Y +k P +k = X P k= (θβ) k E Λ,k Z +k Y +k P +k P +k LHS Þrs bp P b X (θβ) k ΛYP + k= (θβ) k ΛYP E bλ,k + Y b +k +( ) P +k b k= RHS nex X P (θβ) k ΛYP + X (θβ) k ΛYP E Λ b,k + bz +k + by +k + bp +k = P X k= k= X P (θβ) k ΛYP + (θβ) k ΛYP E bλ,k + Z b +k + Y b +k + P +k b k= k= hence: bp bp P k= (θβ)k = P k= (θβ)k E b P +k + bz +k =( θβ) P k= (θβ)k E b P +k + bz +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 θβ o omorrow, θ 2 β 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 θβ, whose inverse premuliplies he summaion - weighs sum up o one -)

10 4 CHAPTER 4. THE DYNAMIC NEW-KEYNESIAN MODEL Bu we know ha: bp θp b = ( θ) P b bp θ bp = ( θ)( θβ) E P b + bz + θβ P b+ + bz + bp θ bp = ( θ)( θβ) P b + bz + θβ E bp + θ bp nex period value of LHS + θ 2 β 2 (...)+... bp bp = ( θ) bp +( θ)( θβ) bp + θβ E bp + θ bp +( θ)( θβ) bz bπ = ( θ) bπ + θβe bπ + +( θ)( θβ) bz bπ = βe bπ + + ( θ)( βθ) θ This equaion is nohing else bu an expecaions augmened Phillips curve, which saes ha inßaion 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. Noice ha µ bπ Z b / β = ( θ) < µ bπ / θ < bz bz he higher β, he higher he weigh o fuure bz s, and he lower oday s elasiciy o curren marginal cos he higher θ, he higher he chance ha I will be suck wih my price for a long period, and he higher he elasiciy of bp o bz. However, few prices will be changed in he aggregae, herefore aggregae inßaion 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 4.5. THE LOG-LINEAR EQUILIBRIUM 4 y = E y + ρ (r E π + ) y = η + ρ z η + η + ρ a π = ( θ)( βθ) βe π + + z + u θ r = φ r r +( φ r )((+φ π ) π + φ z z )+e Remark The linearized equaions are in he Malab Þle, 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 Y = C; anoher deþnes λ as he marginal uiliy of consumpion. I is someimes convenien o call y n a new variable ha deþnes he equilibrium level of oupu (he naural oupu) ha would prevail under compleely ßexible prices (θ =). This way z caninfacbeeliminaed. Infac,IfÞrms were able o adjus prices opimally each period, z =(since ( θ)( βθ) ) and we would be able o deþne he ßexible price θ equilibrium values for real ineres rae and oupu, which we call heir naural raes: r n r n = ρη η + ρ (a E a + ) y n = η η + ρ a We can hen derive an expression for x as a funcion of he gap beween ßexible price and sicky price equilibrium, ha is: x =(η + ρ ) (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 onechnology. Wihhisconvenion,hedynamicnew-keynesianmodelcanberewrienas: y = E y + σ (r E π + ) (a) π = λ (y y n )+βe π + + u (b) r = φ r r +( φ r ) (( + φ π ) π + φ x (y y n )) + e (c) where λ =(η + ρ ) ( θ)( βθ) θ,σ=/ρ. Some auhors have also posulaed cos push shocks u, ha push inßaion up. Some auhors refer o he sysem made by (), (2), (3) as he benchmark dynamic-new keynesian model.

12 42 CHAPTER 4. THE DYNAMIC NEW-KEYNESIAN MODEL TECNO SHOCK.5.5 Y R X π INFLATION SHOCK MONETARY SHOCK years years years years Figure 4.: Simulaions from dnwk.m, ßexible (riangles) versus sicky price (circles) model 4.6 The dynamic effecs 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 ßexible prices versus sicky prices. I was generaed wih dnwk.m Technology. Following a rise in echnology, marginal coss fall. Since no all prices are free o fall immediaely, markups will rise. A fracion θ of ßex price Þrms will lower heir prices and hire more facors of producion. A fracion θ of Þxed price Þrms will be unable o lower heir prices and o increase heir sales and herefore will hire less facors of producion. Producion rises less han wih ßexible prices ρ a =.5;ρ e =.;ρ u =.;β =.99; η =.5;θ =.75 and.; X =.;ρ =; φ r =.8;φ π = 2;φ x =. ;

13 4.6. THE DYNAMIC EFFECTS OF TECHNOLOGY AND MONETARY SHOCKS 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 efficien resource allocaion fails o hold. Moneary A moneary conracion leads o drop in oupu, rise in he nominal ineres rae, fall in inßaion, and a rise in he oupu gap. These predicions, which are qualiaively in line wih he VAR evidence, are hard o obain in he ßexible price model. Inßaion Inßaion shocks are imporan in his seup because hey generae a rade-off beween oupu gap versus inßaion sabilizaion.