Graduate Macro Theory II: A New Keynesian Model with Both Price and Wage Stickiness

Transcription

1 Graduae Macro Theory II: A New Keynesian Model wih Boh Price and Wage Sickiness Eric Sims Universiy of Nore Dame Spring 27 Inroducion This se of noes augmens he basic NK model o include nominal wage rigidiy. Wage rigidiy is inroduced in an analogous way o price rigidiy via he Calvo 983 saggered pricing assumpion, which faciliaes aggregaion. As wih price-seing, o ge wage-seing we need o inroduce some kind of monopoly power in wage-seing. To do his we assume ha households supply differeniaed labor. This imperfec subsiuabiliy beween ypes of labor gives hem some marke power, and allows us o hink abou he consequences of wage sickiness. 2 Producion The producion side of he economy is basically idenical o wha we had in he basic New Keynesian model, and as such we discuss i firs. Producion is spli ino wo secors: a represenaive compeiive final goods firm, and a coninuum of monopolisically compeiive inermediae goods firms who have pricing power bu are subjec o price sickiness via he Calvo 983 assumpion. 2. Final Goods Secor The final oupu good is a CES aggregae of a coninuum of inermediaes: Y = Y j ɛp ɛp ɛp ɛp Here ɛ p >. I index i by p because we ll have a similar parameer a play when i comes o wage sickiness. Profi maximizaion by he final goods firm yields a downward-sloping demand curve for each inermediae: P j ɛp Y j = Y 2

2 This says ha he relaive demand for he j h inermediae is a funcion of is relaive price, wih ɛ p he price elasiciy of demand. The price index derived from he definiion of nominal oupu as he sum of prices imes quaniies of inermediaes can be seen o be: = ɛp j ɛp dj Inermediae Producers A ypical inermediae producers produces oupu according o a consan reurns o scale echnology in labor, wih a common produciviy shock, A : Y j = A N j 4 Inermediae producers face a common wage. They are no freely able o adjus price so as o maximize profi each period, bu will always ac o minimize cos. The cos minimizaion problem is o minimize oal cos subjec o he consrain of producing enough o mee demand: min N j W N j s.. A Lagrangian is: P j ɛp A N j Y P j ɛp L = W N j + ϕ j A N j Y The FOC is: Or: L N j = W = ϕ ja ϕ = W A 5 Here I have dropped he j reference: marginal cos ϕ is equal o he wage divided by produciviy, boh of which are common o all inermediae goods firms. Real flow profi for inermediae producer j is: Π j = j Y j W N j 2

3 From 5, we know W = ϕ A. Plugging his ino he expression for profis, we ge: Π j = j Y j mc Y j Where I have defined mc ϕ as real marginal cos. Firms are no freely able o adjus price each period. In paricular, each period here is a fixed probabiliy of φ p ha a firm can adjus is price. This means ha he probabiliy a firm will be suck wih a price one period is φ p, for wo periods is φ 2 p, and so on. Consider he pricing problem of a firm given he opporuniy o adjus is price in a given period. Since here is a chance ha he firm will ge suck wih is price for muliple periods, he pricing problem becomes dynamic. Firms will discoun profis s periods ino he fuure by M +s φ s p, where M +s = β s u C +s u C is he sochasic discoun facor. The dynamic problem can be wrien: max j E s= βφ p s u C +s u C j +s P j ɛp P j ɛp Y +s mc +s Y +s +s +s Here I have imposed ha oupu will equal demand. Muliplying ou, we ge: max j E s= βφ p s u C +s u C The firs order condiion can be wrien: j ɛp P ɛp +s Y +s mc +s j ɛp P ɛp +s Y +s ɛ p j ɛp E Simplifying: s= βφ p s u C +s P ɛp +s Y +s+ɛ p j ɛp E j = ɛ p ɛ p E s= E s= βφ p s u C +s mc +s P ɛp +s Y +s s= βφ p s u C +s P ɛp +s Y +s βφ p s u C +s mc +s P ɛp +s Y +s = Firs, noe ha since nohing on he righ hand side depends on j, all updaing firms will updae o he same rese price, call i P #. We can wrie he expression more compacly as: Here: X, P # = ɛ p 6 ɛ p X 2, X, = u C mc P ɛp Y + φ p βe X,+ 7 3

4 X 2, = u C P ɛp Y + φ p βe X 2,+ 8 If φ p =, hen he righ hand side would reduce o mc = ϕ. In his case, he opimal price ɛ would be a fixed markup, p ɛ, over nominal marginal cos, ϕ p. 3 Households The new acion relaed o wage sickiness is on he household side. To inroduce wage sickiness in an analogous way o price sickiness, we need households o supply differeniaed labor inpu, which gives hem some pricing power in seing heir own wage. In a similar way o he final goods firm, we inroduce he concep of a labor packer or union, if you like which combines differen ypes of labor ino a composie labor good ha i hen leases o firms a wage rae W. We firs consider he problem of he compeiive labor packing firm, and hen he problem of he household. 3. Labor Packer Toal labor inpu is equal o: N = ɛw N l ɛw ɛw ɛw dl Here ɛ w >, and l indexes he differeniaed labor inpus, which populae he uni inerval. The profi maximizaion problem of he compeiive labor packer is: max N l W ɛw N l ɛw ɛw ɛw dl W ln ldl The firs order condiion for he choice of labor of variey l is: W ɛ w ɛ w This can be simplified somewha: ɛw N l ɛw ɛw dl ɛw ɛ w ɛ w N l ɛw ɛw = W l 9 Or: Or: N l ɛw N l N l ɛw ɛw ɛw dl ɛw N l ɛw ɛw ɛw dl = W l W W l ɛw = W W l ɛw N l = N W 4

5 In a way exacly analogous o inermediae goods, he relaive demand for labor of ype l is a funcion of is relaive wage, wih elasiciy ɛ w. We can derive an aggregae wage index in a similar way o above, by defining: Or: So: W N = W ln ldl = W ɛw = W l ɛw W ɛw N dl W l ɛw dl W = ɛw W l ɛw dl 3.2 Households Households are heerogenous and are indexed by l,, supplying differeniaed labor inpu o he labor packer above. I m going o assume ha preferences are addiively separable in consumpion and labor, which urns ou o be somewha imporan. If wages are subjec o fricions like he Calvo 983 pricing fricion, households will charge differen wages, meaning hey will work differen hours, meaning hey will have differen incomes and herefore differen consumpion and bond-holding decision. Erceg, Henderson, and Levin 2, JME show ha if here exis sae coningen claims ha insure households agains idiosyncraic wage risk, and if preferences are separable in consumpion and leisure, households will be idenical in heir choice of consumpion and bond-holdings, and will only differ in he wage hey charge and labor supply. As such, in he noaion below, I will suppress dependence on l for consumpion and bonds, bu leave i for wages and labor inpu. I also absrac from money alogeher, noing ha I could include real balances as a separable argumen in he uiliy funcion wihou any effecs on he res of he model. The household problem is: max C,N l,w l,b + E = C β σ σ ψ N l +η + η s.. C + B + W ln l + Π + + i B W l ɛw N l = N W is he nominal price of goods, Π is nominal profi disribued from firms, B is he nominal sock of bonds which pay off in period, which pay he nominal ineres rae known in period 5

6 . Imposing ha labor supply exacly equal demand, which allows me o swich noaion from choosing N l o insead choosing W l, a Lagrangian is: L = E β = C σ σ ψ Wl W ɛw N +η + η ɛw W l + λ W l N + Π + + i B C B + W Le s ake he FOC wih respec o C and B +. L = C σ = λ C Combining hese, we ge: L B + = λ + βe λ + + i C σ This is he sandard Euler equaion for bonds. Now, le s hink abou wage seing. a choice variable, insead wriing he problem as choosing W l. = βe C σ + + i + 2 In wriing he Lagrangian, I have eliminaed N l as As wih prices, assume ha households are no freely able o choose heir wage each period. In paricular, each period hey face he probabiliy of being able o adjus heir wage. Wih probabiliy φ w hey are suck wih a wage for one period, φ 2 w for wo periods, and so on. Before proceeding, le s re-wrie he problem in erms of choosing he real wage insead of he nominal wage. The reason we may wan o do his is ha, depending on he moneary policy rule, inflaion could be non-saionary, which would make nominal wages non-saionary, bu real wages saionary. Define he real wage a household charges as: And similarly for he aggregae real wage: w l = W l w = W Since boh of hese real wages are divided by he same price level, he relaive demand for labor of variey l can be wrien eiher in erms of he raio of nominal wages or he raio of real wages, as hese are equivalen. Now, le s consider he problem of a household who can updae is nominal wage in period. The probabiliy ha nominal wage will sill be operaive in period + s is φ s w. The real wage a household charges in period + s if i is suck wih he nominal wage i choose in period is: 6

7 w +s l = W l +s This can be wrien in erms of he period real wage as: w +s l = W l Define Π,+s = +s of period-over-period gross inflaion. Define π = we have: Π,+s = +s as he gross inflaion beween and + s. This is jus equal o he produc m= P as he period-over-period ne inflaion, s + π +m = s = +s + +s This means ha he real wage a household wih a suck nominal wage will charge in period + s can be wrien: Where w is he real wage chosen in period. w +s l = w lπ,+s Now, when choosing w l, households will discoun he fuure no jus by β s bu by φ s w as well, since he laer is he probabiliy ha a household will be suck wih ha wage in period + s. Reproducing jus he pars of he Lagrangian ha relaed o he choice of labor, we have: ɛw+η w lπ L,+s = E βφ w s ψ w +s N +η +s + η s= + λ +s +s w lπ,+s w lπ,+s w +s ɛw N +s Noe ha he muliplier, λ +s, ges muliplied by +s because I m wriing he wage in real erms here so I m de-faco muliplying and dividing by +s. By muliplying ou, his can be re-wrien: L = E βφ w s ψ w l ɛw+η w ɛw+η +s + η s= Π ɛw+η,+s N +η +s + λ +s +s w l ɛw w+s ɛw Πɛw,+s N +s The firs order condiion is: 7

8 Simplifying: L w l = ɛ ww l ɛw+η E s=... ɛ w w l ɛw E s= βφ w s ψw ɛw+η +s Π ɛw+η,+s N +η +s +... βφ w s λ +s +s w ɛw +s Πɛw,+s N +s = ɛ w w l ɛw+η E Or: s= βφ w s ψw ɛw+η +s Π ɛw+η,+s N +η +s = ɛ w w l ɛw E w #,+ɛwη = ɛ w ɛ w E s= E s= βφ w s ψw ɛw+η +s Π ɛw+η,+s N +η +s βφ w s λ +s +s w ɛw +s Πɛw,+s N +s s= βφ w s λ +s +s w ɛw +s Πɛw,+s N +s Above, I have goen rid of he dependence on he l index, because everyhing on he righ hand side is independen of l, meaning ha all updaing households will updae o he same wage, which I call w # or he rese wage. This can be wrien more compacly as: 3 Where: w #,+ɛwη = ɛ w ɛ w H, H 2, 4 H, = ψw ɛw+η N +η + βφ w E + π + ɛw+η H,+ 5 H 2, = C σ w ɛw N + βφ w E + π + ɛw H 2,+ 6 These lines follow because Π, =, and Π,+ = + π +, so he Π,+s is effecively like an addiional par of he discoun facor, and λ = C σ. 3.3 Wha if wages were flexible? This FOC for labor inpu looks complicaed, and in paricular looks differen han a normal saic FOC for labor. To see ha i s no so crazy, consider he case of wage flexibiliy, in which φ w =. Then he FOC would break down o: w #,+ɛwη = ɛ w ψw ɛw+η N +η ɛ w C σ w ɛw N 8

9 If φ w =, hen all firms updae, so he rese wage is equal o he acual real wage: w # = w. This means: Or: w +ɛwη = ɛ w ψn η wɛwη ɛ w C σ w = ɛ w ψn η ɛ w C σ ɛ Since ɛ w >, we have w ɛ w >. So wha his says is ha wage is a markup over he marginal rae of subsiuion beween labor and consumpion ψn η is he MRS. If ɛ C σ w, his would be exacly he FOC ha we had in he flexible wage case. 4 Equilibrium and Aggregaion Assume ha he cenral bank ses ineres raes according o a Taylor Rule. In he Taylor rule I arge only inflaion, bu i would be sraighforward o also arge oupu, he oupu gap, or oupu growh. As long as households ge uiliy from real balances in an addiively separable way, his will deermine he price level and we can ignore money: i = ρ i i + ρ i i + φ π π π + ε i, 7 Where again variables wihou ime subscrips denoe seady sae values. ε i, is a moneary policy shock. Produciviy follows an AR in he log: ln A = ρ a ln A + ε a, 8 In equilibrium, bond-holding is always zero: B =. Using his, he household budge consrain can be wrien in real erms: Inegraing over l: C = W l N l + Π 9 C = W l N ldl + Π Real dividends received by he household are jus he sum of real profis from inermediae goods firms: This can be wrien: Π = P j Y j W N j dj 9

10 Π = j Y j w N jdj Where above I used he definiion ha w W. Now, marke-clearing requires ha he sum of labor used by firms equals he oal labor supplied by he labor packer, so N jdj = N. Hence: Π = j Y jdj w N Plug his ino he inegraed household budge consrain: C = W l N ldl + Now plug in he demand for labor of ype l: j Y jdj w N Simplify: C = W l W l ɛw N dl + W j Y jdj w N C = W ɛw N W l ɛw dl + j Y jdj w N Now, using he aggregae nominal wage index, we know: W l ɛw dl = W ɛw. Making his subsiuion: Or: C = W ɛw N W ɛw + j Y jdj w N Since w = W, we mus have: C = W N + C = j Y jdj w N j Y jdj 2 In oher words, consumpion mus equal he sum of real quaniies of inermediaes. Now, plug in he demand curve for inermediae variey j: C = j Take suff ou of he inegral where possible: P j ɛp Y dj C = P ɛp Y j ɛp dj

11 Now, from he definiion of he aggregae price level, we have: means he erms involving cancel, so we re lef wih: j ɛp = P ɛp. This C = Y 2 Now, wha is Y? From he demand for inermediae variey j, we have: P j ɛp Y j = Y Using he producion funcion for each inermediae, his is: Inegrae over j: P j ɛp A N j = Y A N jdj = P j ɛp Y dj Take suff ou of he inegral, wih he excepion of he price level on he righ hand side: Now define a new variable, v p, as: P j ɛp A N jdj = Y dj v p = P j ɛp dj 22 This is a measure of price dispersion. If here were no pricing fricions, all firms would charge he same price, and v p =. If prices are differen, one can show ha his expression is bound from below by uniy. Using he definiion of aggregae labor inpu, we can herefore wrie: Y = A N v p 23 This is he aggregae producion funcion Since v p, price dispersion resuls in an oupu loss you produce less oupu han you would given A and aggregae labor inpu if prices are disperse. Since I ve wrien he firs order condiions for labor in erms of he real wage, le s re-wrie he aggregae nominal wage index in erms of real wages. Divide boh sides by P ɛw : W ɛw W l ɛw = dl Or:

12 w ɛw = The full se of equilibrium condiions can hen be characerized by: w l ɛw dl 24 C σ = βe C σ + + i + 25 w #,+ɛwη = ɛ w ɛ w H, H 2, 26 H, = ψw ɛw+η N +η + βφ w E + π + ɛw+η H,+ 27 H 2, = C σ w ɛw N + βφ w E + π + ɛw H 2,+ 28 mc = w A 29 C = Y 3 Y = A N v p 3 v p = P j ɛp dj 32 w ɛw = P ɛp = w l ɛw dl 33 j ɛp dj 34 X, P # = ɛ p 35 ɛ p X 2, X, = C σ mc P ɛp Y + φ p βe X,+ 36 X 2, = C σ P ɛp Y + φ p βe X 2,+ 37 i = ρ i i + ρ i i + φ π π π + ε i, 38 ln A = ρ a ln A + ε a, 39 π = This is sixeen equaions in sixeen aggregae variables: C, i,, w #, H,, H 2,, w, N, π, mc, A, Y, v p, P #, X,, X 2,. 4. Re-Wriing Equilibrium Condiions There are wo issues wih how I ve wrien hese condiions. 4 Firs, I haven goen rid of he heerogeneiy I sill have j and l indexes showing up. Second, I have he price level showing up, 2

13 which, as I menioned above, may no be saionary. Hence, I wan o re-wrie hese condiions i only in erms of inflaion, eliminaing he price level; and ii geing rid of he heerogeneiy, which he Calvo 983 assumpion allows me o do. The Euler equaion can be rivially re-wrien in erms of inflaion as: C σ = βe C σ + + i + π + 4 Le s look a he expressions for he price level and he real wage. The expression for he price level is: P ɛp = j ɛp dj Now, a fracion φ p of hese firms will updae heir price o he same rese price, P #. The oher fracion φ p will charge he price hey charged in he previous period. This means we can break up he inegral on he righ hand side as: This can be wrien: φp P ɛp = P #, ɛp dj + j ɛp dj φ p P ɛp = φ p P #, ɛp + j ɛp dj φ p Now, here s he beauy of he Calvo assumpion. Because he firms who ge o updae are randomly chosen, and because here are a large number coninuum of firms, he inegral sum of individual prices over some subse of he uni inerval will simply be proporional o he inegral over he enire uni inerval, where he proporion is equal o he subse of he uni inerval over which he inegral is aken. This means: φ p j ɛp dj = φ p j ɛp dj = φ p P ɛp This means ha he aggregae price level raised o ɛ p is a convex combinaion of he rese price and lagged price level raised o he same power. So: P ɛp = φ p P #, ɛp + φ p P ɛp In oher words, we ve goen rid of he heerogeneiy. The Calvo assumpion allows us o inegrae ou he heerogeneiy and no worry abou keeping rack of wha each firm is doing from he perspecive of looking a he behavior of aggregaes. Now, we sill have he issue here ha we are wrien in erms of he price level, no inflaion. To ge i in erms of inflaion, divide boh sides by P ɛp, and define π# = P # as rese price inflaion: 3

14 + π ɛp = φ p + π # ɛp + φ p 42 We can do exacly he analogous hing for wages. The aggregae real wage index is: w ɛw = w l ɛw dl Since of households will updae he same rese wage, and φ w will be suck wih las period s nominal wage, his is: w ɛw = w #, ɛw + φ w W ɛw dl Noe ha I have wrien his in erms of nominal wages in erms of he non-updaed wages. We can re-wrie in erms of real wages as: w ɛw = w #, ɛw + φ w W ɛw ɛw P dl Wrien in erms of inflaion, and aking i ou of he inegral, his is jus: w ɛw = w #, ɛw + + π ɛw φ w w l ɛw dl Again, he Calvo assumpion allows us o ge rid of he inegral on he righ hand side, which will jus be proporional o las period s aggregae real wage. So we re lef wih: w ɛw = w #, ɛw + φ w + π ɛw w ɛw 43 We can also use he Calvo assumpion o break up he price dispersion erm, by again noing ha φ p of firms will updae o he same price, and φ p firms will be suck wih las period s price. Hence: v p = φp P # ɛp dj + φ p P j ɛp dj This can be wrien in erms of inflaion by muliplying and dividing by powers of where necessary: φp v p = P # ɛp ɛp P dj + We can ake suff ou of he inegral: φ p P j ɛp ɛp P dj v p = φ p + π # ɛp + π ɛp + + π ɛp φ p P j ɛp dj 4

15 By he same Calvo logic, he erm inside he inegral is jus going o be proporional o v p. This means we can wrie he price dispersion erm as: v p = φ p + π # ɛp + π ɛp + + π ɛp φ p v p 44 In oher words, we jus have o keep rack of v p, no he individual prices. Now, we need o adjus he rese price expression. Firs, define wo new auxiliary variables as follows: x, X, P ɛp x 2, X 2, P ɛp Dividing boh sides of he rese price expressions by he appropriae power of, we have: x, = C σ x 2, = C σ mc Y + φ p βe X,+ P ɛp Y + φ p βe X 2,+ P ɛp Muliplying and dividing he + erms by he appropriae power of +, we have: Or, in erms of inflaion: x, = C σ x 2, = C σ mc Y + φ p βe X,+ P ɛp + Y + φ p βe X 2,+ P ɛp + ɛp P+ ɛp P+ x, = C σ mc Y + φ p βe + π + ɛp x,+ 45 x 2, = C σ Y + φ p βe + π + ɛp x 2,+ 46 Now, in erms of he rese price expression, since we divided X, by P ɛp and divided X 2, by P ɛp, we de-faco muliply he raio of X, X 2, by P. Hence, o keep equaliy, we need o muliply he righ hand side by. Hence, he rese price expression can now be wrien: P # = ɛ p ɛ p Now, simply divide boh sides by o have everyhing in erms of inflaion raes: x, x 2, + π # = ɛ p ɛ p + π x, x 2, 47 5

16 The full se of equilibrium condiions can now be expressed: C σ = βe C σ + + i + π + 48 w #,+ɛwη = ɛ w ɛ w H, H 2, 49 H, = ψw ɛw+η N +η + βφ w E + π + ɛw+η H,+ 5 H 2, = C σ w ɛw N + βφ w E + π + ɛw H 2,+ 5 mc = w A 52 C = Y 53 Y = A N v p 54 v p = φ p + π # ɛp + π ɛp + + π ɛp φ p v p 55 w ɛw = w #, ɛw + φ w + π ɛw w ɛw 56 + π ɛp = φ p + π # ɛp + φ p 57 + π # = ɛ p ɛ p + π x, x 2, 58 x, = C σ mc Y + φ p βe + π + ɛp x,+ 59 x 2, = C σ Y + φ p βe + π + ɛp x 2,+ 6 i = ρ i i + ρ i i + φ π π π + ε i, 6 ln A = ρ a ln A + ε a, 62 This is now fifeen equaions in fifeen variables, where I have eliminaed as a variable, replaced P # wih π #, and replaced X, and X 2, wih x, and x 2,. 5 Seady Sae In he non-sochasic seady sae, A =. Seady sae inflaion will be equal o arge. We can solve for seady sae rese price inflaion as: + π + π # ɛ p φ p = φ p ɛp 63 Here we see ha if π =, hen π # = as well. Seady sae price dispersion is: v p = φ p +π +π # ɛp + π ɛp φ 64 6

17 From his, we can again see ha if π = π # =, hen v p =. The seady sae nominal ineres rae is: The seady sae auxiliary pricing variables are: + i = + π 65 β This means he raio is: x = x 2 = Y σ mc φ p β + π ɛp 66 Y σ φ p β + π ɛp 67 x = mc φ pβ + π ɛp x 2 φ p β + π ɛp Hence, we can solve for seady sae marginal cos as: mc = ɛ p ɛ p φ p β + π ɛp + π # φ p β + π ɛp + π Here again, we see ha if π = π # =, hen mc = ɛp ɛ p, which is he desired flexible price markup. Le s solve for he opimal rese wage in erms of he seady sae real wage: w # = φw + π ɛw 68 ɛw w 69 This says ha he rese wage is proporional o he seady sae wage. Noe ha, if φ w = wages were flexible, we would have w # = w. Le s now solve for he seady saes of he auxiliary variables relaed o wage-seing. We have: The raio is jus: H = H 2 = ψw ɛw+η N +η β + π ɛw+η 7 Y σ w ɛw N β + π ɛw 7 H = ψw ɛwη Y σ N η β + π ɛw H 2 β + π ɛw+η Plug his ino he FOC for labor: w #,+ɛwη = ɛ w ɛ w ψwɛwη Y σ N η β + π ɛw β + π ɛw+η 7

18 Now, as an insrucive exercise, suppose ha φ w =, so ha wages were flexible. This would mean w # = w. Combining hese, we d have: This is insrucive. ɛ w Y σ w = ψn η ɛ w If ɛ w, his would be he same saic FOC ha we ve had before: he marginal disuiliy of labor mus equal he marginal uiliy of consumpion Y σ here, since C = Y imes he real wage. If you defined he MRS marginal rae of subsiuion beween labor and consumpion as ψn η Y σ, hen you could re-wrie his as: w = ɛ w ɛ w MRS In oher words, households se he wage as a markup over he marginal rae of subsiuion, in an analogous way o how firms se price as a markup over marginal cos. have: Now, go back o our earlier expression from he FOC for labor. Eliminaing he rese wage, we φw + π ɛw +ɛwη ɛw w +ɛwη = ɛ w ɛ w ψwɛwη Y σ N η β + π ɛw β + π ɛw+η Simplify: N η = ɛ w ɛ w ψ Y σ w φw + π ɛw +ɛwη ɛw β + π ɛw+η β + π ɛw Now, wha does his do for us? Well, we know ha w = mc, and we know ha N = Y v p. Plugging hese in: N η = ɛ w ɛ w ψ N σ v p σ mc φw + π ɛw +ɛwη ɛw β + π ɛw+η β + π ɛw Now we can solve for N: N = ɛ w ɛ w ψ vp σ mc φw + π ɛw +ɛwη ɛw β + π ɛw+η β + π ɛw σ+η 72 Once we know N, we know Y as well. 8

19 6 Quaniaive Analysis I solve he model quaniaively using a firs order approximaion in Dynare. I use he following parameer values: ɛ p = ɛ w =, φ p = φ w =.75, β =.99, σ =, ψ =, η =, ρ a =.95, ρ i =.8, φ π =.5, and I se he sandard deviaions of he produciviy and moneary policy shocks o. and.25, respecively. Below are he impulse responses o a produciviy shock. Oupu rises, by an amoun fairly close o is flexible price level, hours decline, he nominal ineres rae declines, he real ineres rae rises, and he real wage rises..5 A.5 Oupu 2 x 3 Hours x 3 Nom. Rae 2 x 3 Inflaion x 3 Real Wage x 3 Real Rae The responses o a moneary policy shock are shown below. Boh he real and nominal ineres raes rise. Oupu, hours, inflaion, and he real wage all decline. 9

20 5 x 3 Oupu x 4 Inflaion x 3 Hours x 4 Real Wage x 3 Nom. Rae x 3 Real Rae 2 2 Wha is he relaive imporance of price and wage sickiness in accouning for his paern of impulse responses? We observe ha wage rigidiy in isolaion acually amplifies he responses of real variables o a produciviy shock, whereas price rigidiy in isolaion dampens hose responses. The responses wih boh price and wage rigidiy are somewhere in beween. In conras, he reacions o he moneary policy shock are prey similar wih eiher wage or price rigidiy. 2

21 .5 A.2 Oupu x 3 Hours x 3 Nom. Rae 5 x 3 Inflaion.5 Real Wage x 3 Real Rae Baseline Only Prices Sicky Only Wages Sicky x 3 Oupu x 3 Inflaion x 3 Hours x 3 Real Wage x 3 Nom. Rae x 3 Real Rae Baseline Only Prices Sicky Only Wages Sicky 2

22 We can undersand he paern of impulse responses by noing ha here are wo monopoly disorions in he model. One relaes o price-seing price would be a fixed markup over marginal cos if prices were flexible and one relaes o wage-seing he real wage would be a fixed markup over he marginal rae of subsiuion. We can hink of price and wage rigidiy causing hese markups o vary endogenously in he shor run in response o shocks. The price markup is jus he negaive of real marginal cos, and he wage markup is he raio of he real wage o he marginal rae of subsiuion. Oupu will respond by less han i would if prices were flexible if eiher of hese markups rise in response o a shock; if eiher markup falls, his is relaively expansionary. In he figure below, I plo he price and wage markups o boh a produciviy and moneary policy shock under hree regimes: one where boh prices and wages are sicky, one where only prices are sicky, and one where only wages are sicky. In response o eiher shock, when only wages are sicky, he price markup is fixed. In conras, when only prices are sicky, he wage markup is fixed. Produciviy Shock 8 x Price Markup 5 x 3 Wage Markup Price Markup.25 Wage Markup Moneary Shock Baseline Only Prices Sicky Only Wages Sicky Above, we see ha boh he price and wage markups go up in response o he moneary policy shock, bu hey go in opposie direcions afer a produciviy shock. A produciviy shock pus upward pressure on real wages and downward pressure on prices. Downward pressure on prices means ha some firms will end up wih prices ha are higher han hey would like, hence, if prices are sicky, he price markup rises, which effecively means he economy is more disored. This is why oupu rises by less han i would if prices were flexible in a sicky price model in response o a produciviy shock. The opposie paern occurs wih wages. Real wages need o rise afer a 22

23 posiive produciviy shock; because some households can adjus heir nominal wages, hey end up wih wage markups ha are oo low. Hence, he aggregae wage markup falls, which means ha he economy is relaively undisored along ha dimension, which is relaively expansionary. This is why, when only wages are sicky, oupu rises by more han i would under flexible prices and wages, because he wage markup ges squeezed. The differenial behavior of he price and wage markups in response o he produciviy shock accouns for why he oupu responses o he shock look so differen when one of he sickiness parameers is urned off. In response o he moneary policy shock, boh he wage and price markups move in he same direcion. The conracionary moneary policy shock pus downward pressure on prices, so some firms end up wih prices ha are oo high relaive o wha hey would opimally like he price markup rises. If prices and wages were boh flexible, here would be no effec on real wages of a moneary policy shock. The downward pressure on prices herefore means ha here is downward pressure on wages. Since some households can adjus heir wages downward, hey end up wih wages ha are oo high, and he economy-wide wage markup rises. The increases in boh he price and wage markups are conracionary in he case of a moneary policy shock. Since he markups behave in he same way, we observe ha here is a much smaller difference in he responses o a moneary policy shock when eiher prices or wages are sicky, relaive o he case of a produciviy shock. 7 Log-Linearizaion Now, le s log-linearize he equilibrium condiions. We are going o do his abou a zero inflaion seady sae, which makes life much easier. Sar wih he Euler equaion, going ahead and imposing he accouning ideniy ha C = Y. We have: σ ln Y = ln β σe ln Y + + i E π + σỹ = σe Ỹ + + ĩ E π + Where Ỹ = Y Y Y, ĩ = i i, and π = π π. In oher words, he variables already in rae form ineres rae and inflaion are expressed as absolue deviaions, and variables no already in rae form as percen log deviaions. We can re-wrie his as: Ỹ = E Ỹ + σ ĩ E π + This is someimes called he New Keynesian IS Curve. Real marginal cos is already log-linear: 73 Log-linearize he producion funcion: mc = w à 74 23

24 Ỹ = Ã + Ñ + ṽ p Now, wha is ṽ p? Le s ake logs and go from here: ln v p = ln φ + π # ɛp + π ɛp + + π ɛp φv p Now, from our discussion above, we know ha v p = when π =. Toally differeniaing: ṽ p = ɛ p φ + π # ɛp + π ɛp π # π # + ɛ p φ + π # ɛp + π ɛp π π ɛ p + π ɛp φv p π π + + π ɛp φv p vp Using now known facs abou he seady sae, his reduces o: ṽ p = ɛ p φ π # + ɛ p φ π + ɛ p φ π + φṽ p This can be wrien: ṽ p = ɛ p φ π # + ɛ p π + φṽ p Now, log-linearize he equaion for he evoluion of inflaion: ɛ p π = ln φ + π # ɛp + φ ɛ p π π = + π ɛp ɛ p φ + π # ɛp π # π # In he las line above, he + π ɛp shows up because he erm inside parenheses is equal o + π ɛp evaluaed in he seady sae, and when aking he derivaive of he log his erm ges invered evaluaed a ha poin. Using facs abou he zero inflaion seady sae, we have: Or: ɛ p π = ɛ p φ π # π = φ π # 75 In oher words, acual inflaion is jus proporional o rese price inflaion, where he consan is equal o he fracion of firms ha are updaing heir prices. This is prey inuiive. Now, use his in he expression for price dispersion: 24

25 ṽ p = ɛ p π φ π # + φṽ p Bu from above, he firs erm drops ou, so we are lef wih: ṽ p = φṽp 76 If we are approximaing abou he zero inflaion seady sae in which v p =, hen we re saring from a posiion in which ṽ p =, so his means ha ṽp = a all imes. In oher words, abou a zero inflaion seady sae, price dispersion is a second order phenomenon, and we can jus ignore i in a firs order approximaion abou a zero inflaion seady sae. Given his, he log-linearized producion funcion is jus: Ỹ = Ã + Ñ 77 Now, le s log-linearize he rese price expression. This is muliplicaive, and so is already in log-linear form. We have: have: π # = π + x, x 2, 78 Now we need o log-linearize he auxiliary variables. Imposing he ideniy ha Y = C, we Toally differeniaing: ln x, = ln Y σ mc + φβe + π + ɛp x,+ x, x x = x σy σ mcy Y + Y σ mc mc + ɛ p φβ + π ɛp x π + π + φβ + π ɛp x,+ x Disribuing he x and muliplying, dividing where necessary o ge in o percen deviaion erms, and making use of he coninued assumpion of linearizaion abou a zero inflaion seady sae, we have: x, = σy σ mc Ỹ + Y σ mc mc + ɛ p φβe π + + φβe x,+ x x Now, wih zero seady sae inflaion, we know ha x = Y σ mc φβ. This simplifies he firs wo erms: x, = σ φβỹ + φβ mc + ɛ p φβe π + + φβe x,

26 Now, log-linearize x 2, : Toally differeniaing: ln x 2, = ln Y σ + φβe + π + ɛp x 2,+ x 2, x 2 x 2 = x 2 σy σ Y Y + ɛ p φβ + π ɛp 2 x 2 π + π + φβ + π ɛp x 2,+ x 2 Disribuing he x 2, muliplying and dividing by appropriae erms, and making use of he fac ha π =, we have: σy σ x 2, = Ỹ + ɛ p φβe π + + φβe x 2,+ x 2 Since x 2 = Y σ φβ, his can be wrien: x 2, = σ φβỹ + ɛ p φβe π + + φβe x 2,+ 8 Now, subracing x 2, from x,, we have: x, x 2, = φβ mc + φβe π + + φβe x,+ x 2,+ From above, we also know ha: x, x 2, = π # π Bu π # = φ π, so we mus also have: Make his subsiuion above: Muliplying hrough: x, x 2, = φ φ π φ φ φ π = φβ mc + φβe π + + φβe φ E π + Or: π = φ φβ φ mc + φβe π + + φβe π + π = φ φβ φ mc + βe π

27 This is he sandard New Keynesian Phillips Curve. presence of wage rigidiy. Is basic srucure is unalered by he Now, le s go log-linearize he wage-seing equaions. Begin by aking logs of he aggregae real wage series: Toally differeniae: ɛ w ln w = ln w #, ɛw + φ w + π ɛw w ɛw ɛ w w w w = ɛ w w #, ɛw w # w ɛw w # + ɛ w φ w w ɛw π π + ɛ w φ w w ɛw w w Above, I have made use of he fac ha we are linearizing abou he poin π =, which simplifies hings a bi. Also, since we are linearizing abou a zero inflaion seady sae, we know from above ha w # = w. Making use of his, we have: Simplifying: ɛ w w = ɛ w w # ɛ w φ w π + ɛ w φ w w w = w # + φ w w π 82 This is prey inuiive. I says ha he curren real wage is a convex combinaion of he rese real wage and las period s real wage, minus an adjusmen for inflaion. The reason he adjusmen for inflaion is because nominal wages are fixed. Now, le s log-linearize he rese wage equaion. Since i is muliplicaive, i is already log-linear: + ɛ w η w # = H, H 2, 83 Now we need o log-linearize he auxiliary wage-seing variables. Le s log: ln H, = ln ψw ɛw+η N +η + βφ w E + π + ɛw+η H,+ H, = H ɛ w + ηψw ɛw+η N +η w w + + ηψw ɛw+η N η N N + ɛ w + ηβφ w H π + π +... In simplifying his, noe ha H = ψwɛw+η N +η φ wβ. Disribuing hings:... +φ w βh,+ H H, = βɛ w + η w + β + ηñ + ɛ w + ηφ w βe π + + φ w βe H,+ 27

28 Now, le s do H 2, : Toally differeniae: ln H 2, = ln Y σ w ɛw N + βφ w E + π + ɛw H 2,+ H 2, = H 2 σy σ w ɛw NY Y + ɛ w Y σ w ɛw Nw w + Y σ w ɛw N N + ɛ w φ w βh 2 π + π +... In simplifying his, noe ha H 2 = Y σ w ɛw N φ wβ. We can now wrie his: + φ w βh 2,+ H 2 H 2, = βσỹ + βɛ w w + βñ + ɛ w φ w βe π + + φ w βe H2,+ Hence, we have: H, H 2, = βɛ w η w + βηñ + βσỹ + φ w β + ɛ w ηe π + + φ w βe H,+ H 2,+ The MRS beween labor and consumpion, as inroduced above, is MRS = ψn η Y σ. In loglinear erms, his is: This means we can wrie: mrs = ηñ + σỹ H, H 2, = βɛ w η w + β mrs + φ w β + ɛ w ηe π + + φ w βe H,+ H 2,+ Le s define µ = mrs w as he gap beween he MRS and he real wage. Playing around: H, H 2, = βɛ w η w + β µ + β w + φ w β + ɛ w ηe π + + φ w βe H,+ H 2,+ Or: H, H 2, = β + ɛ w η w + β µ + φ w β + ɛ w ηe π + + φ w βe H,+ H 2,+ Now, from above we know ha we can wrie he rese wage as: w # = w φ w w + φ w π 28

29 And we also know ha: Combining hese expressions, we have: H, H 2, = + ɛ w η w # H, H 2, = + ɛ wη w + ɛ wηφ w w + + ɛ wηφ w π I is helpful o re-wrie his in erms of he nominal wage: W = w +. Doing so, we have: H, H 2, = + ɛ wη W + ɛ wηφ w W + + ɛ wηφ w π Now, define π w = W W as nominal wage inflaion. We can furher simplify as: H, H 2, = + ɛ wη W φ P φ w W + φ w P + φ w π w H, H 2, = + ɛ wη W φ W + W P P P + φ w π w In erms of he real wage again, his can be re-wrien: H, H 2, = + ɛ wη π w + + ɛ w η W + ɛ w η H, H 2, = + ɛ wη π w + + ɛ w η w + ɛ w η π Now, combine his wih our earlier expression for he difference beween he auxiliary variables. We have: ϕ w π w + w π = β w + β + ɛ w η µ + φ w βe π + + φ wβ + ɛ w η E H,+ E H2,+ have: Plugging in he same expression for he expeced fuure difference in he auxiliary variables, we Simplifying: π w + w π = β w + β ϕ w + ɛ w η µ + φ w βe π φ wβ + ɛw η E π + w + + ɛ w η w + ɛ w ηe π + + ɛ w η 29

30 Simplifying: ϕ w π w + w π = β w + β + ɛ w η µ + φ w βe π φ w β E π w + + φ w β w φ w βe π + Re-arranging: π w + w π = w + β ϕ w + ɛ w η µ + φ wβ E π + w π w = w w + π + β + ɛ w η µ + φ wβ E π + w Now, noe ha w w + π = π w. Hence, we have: Or: π w = β + ɛ w η µ + φ wβ E π w + And finally: φ w π w = β + ɛ w η µ + φ wβ E π w + π w = β µ + βẽπ+ w 84 φ w + ɛ w η This is he wage Phillips Curve. I looks almos he same as he price Phillips Curve, bu here is an exra erms + ɛ w η in he denominaor. Since ɛ w η >, his means ha he wage Phillips Curve is always flaer han he price Phillips Curve for equal values of he Calvo parameers, φ p and φ w. Also, differenly han for prices, he elasiciy parameer ɛ w shows up in he log-linearized Phillips Curve for wages, whereas i does no for prices. The full se of log-linearized firs order condiions can be wrien: Ỹ = E Ỹ + σ ĩ E π + 85 mc = w à 86 Ỹ = à + Ñ 87 π = φ p φ p β φ p mc + βe π + 88 π w = β µ + βẽπ+ w 89 φ w + ɛ w η µ = mrs w 9 3

31 mrs = ηñ + σỹ 9 π w = w w + π 92 ĩ = ρ i ĩ + ρ i φ π π + ε i, 93 à = ρ a à + ε a, 94 This is en equaions in en unknowns Ỹ, Ñ, mc, ĩ, mrs, µ, w, Ã, π, π w. 7. Gap Formulaion As in he simpler NK model, here are some redundan variables here ha could be eliminaed, and we migh like o wrie he Phillips Curves for prices and inflaion in erms of gaps. As we did earlier, le s define variables wih a superscrip f as he flexible price/wage variables: he values of endogenous variables which would obain in he absence of boh price and wage sickiness. If boh prices and wages were flexible, we d have µ = mc =. This would imply: w f = ηñ f Ỹ f w f = à + σỹ f = à + Ñ f Plugging he firs and hird of he above expressions ino he middle, we ge: Simplifying: f à = η Ỹ à + σỹ f Ỹ f = + η σ + η à 95 Unsurprisingly, his is he same log-linearized expression for he naural rae of oupu as we had before. Le s play around wih he definiion of µ a lile bi: µ = ηñ + σỹ w µ = η Ỹ à + σỹ w µ = σ + ηỹ ηã w Now, add and subrac à from he righ hand side: µ = σ + ηỹ + ηã + à w 3

32 Simplifying: µ = σ + η Ỹ + η σ + η à w à Now, define he real wage gap as X w = w Ã, since we know ha he flexible price real wage would jus be w f = Ã. The oupu gap, X = Ỹ Ỹ f, is he same as before. This means we can wrie his expression as: We can hen plug his ino he wage Phillips Curve o ge: µ = σ + η X X w 96 π w = β σ + η φ w + ɛ w η X β X w + βe π + w 97 φ w + ɛ w η The price Phillips Curve can be wrien in erms of he real wage gap, since real marginal cos is he same hing as he real wage gap: π = φ p φ p β φ p Xw + βe π + 98 The Euler/IS equaion can be wrien in erms of he oupu gap: X = E X+ σ ĩ E π + r f 99 Where: r f = σ + η σ + η ρ a à We can re-wrie he wage inflaion evoluion equaion in erms of he real wage gap as well: π w = w à + à w + à à + π π w = X w X w + à à + π We can wrie he full sysem of equilibrium condiions as: X = E X+ σ ĩ E π + r f π = φ p φ p β φ p Xw + βe π + 2 π w = β σ + η φ w + ɛ w η X β X w + βe π + w 3 φ w + ɛ w η r f = σ + η σ + η ρ a à 4 32

33 π w = X w X w + à à + π 5 ĩ = ρ i ĩ + ρ i φ π π + ε i, 6 à = ρ a à + ε a, 7 There is no way o wrie he Price Phillips Curve solely in erms of he oupu gap when wages are sicky. In he model wih jus price sickiness, we were able o wrie marginal cos in erms of he oupu gap by eliminaing he real wage using he saic firs order condiion for labor supply, so we could wrie marginal cos jus in erms of Ỹ and Ã. Here ha isn sraighforward since he firs order condiion for labor supply is subsanially more complicaed. 7.2 Opimal Moneary Policy As in he model wih jus price sickiness, i is possible o derive a welfare loss funcion from aking a second order approximaion o he household s value funcion while using a firs order approximaion o he equilibrium condiions. The loss funcion now depends on he squared values of he oupu gap, price inflaion, and wage inflaion: These coefficiens are given by: L = λ p ɛ p X2 + π 2 + λ p λ w ɛ w ɛ p π w 2 λ p = φ p φ p β φ p σ + η λ w = β φ w + ɛ w η Hence, he relaive weigh on he oupu gap is he same as in he simpler model. λ w is jus he coefficien on he real wage gap in he wage Phillips Curve. The relaive weigh on wage inflaion depends on i he relaive coefficiens λ p and λ w, and ii he relaive elasiciies of goods and labor demand, ɛ p and ɛ w. Why is wage inflaion an argumen in he loss funcion? I shows up for an analogous reason o why price inflaion shows up. To hink abou aggregae welfare, we need o come up wih a social welfare funcion since here is no a represenaive agen in his model. The easies aggregae welfare funcion is he uiliarian one in which we sum up welfare of individual households. Individual welfare is: Define aggregae welfare as: V l = C σ ψ N l +η + βe V + l σ + η 33

34 W = C σ ψ σ + η Now, noe ha he demand for labor of variey l: W = V ldl N l +η dl + βe W + W l ɛw N l = N W Plug his in o he expression for welfare above: Define: W = C σ ψ σ + η W l W ɛw+η N +η dl + βe W + v w = W l W ɛw+η This is a measure of wage dispersion, and i is bound from below by. This can be wrien recursively if we wan as we previously did for prices, using he assumpions of he Calvo mechanism. Aggregae welfare hen becomes: W = C σ ψ σ + η vw N +η + βe W + 8 Hence, he reason wage inflaion maers is ha wage dispersion effecively drives a wedge beween labor supplied and labor used in producion: if v w >, here is some labor los in he process, in a way analogous o how price dispersion resuls in some los oupu. Going back o he approximaed loss funcion, he inuiion for he relaive weigh on wage inflaion is fairly inuiive. Price or wage inflaion are cosly o he exen o which prices or wages are sicky: if aggregae prices or wages move around, and prices are sicky, his induces price or wage dispersion. The bigger is ε p, he more cosly is price dispersion he lower is he weigh on wage inflaion; he bigger is ε w, he more cosly is wage dispersion he bigger is he weigh on wage inflaion. The sickier are prices, he smaller is λ p, and hence he smaller is he relaive weigh on wage inflaion he bigger is he relaive weigh on price inflaion. Conversely, he sickier are wages, he smaller λ w is, and he bigger he relaive weigh on inflaion. I is insrucive o hink abou wha he relaive weigh on wage inflaion ough o look like by considering some numeric values. Suppose ha φ p = φ w =.75, and ɛ p = ɛ w =, wih σ = η =, and β =.99. We ge λ p =.77, bu λ w =.78. This means ha he relaive weigh on wage inflaion is 22 wage inflaion is 22 imes more imporan han price inflaion. Wha really drives his is ha he wage Phillips Curve is much flaer han he price Phillips Curve because of he presence of ɛ w in he denominaor of he slope coefficien. 34

35 Before doing anyhing quaniaively, i is useful o sop and hink for a minue. In he basic NK model, i was possible o compleely sabilize boh inflaion and he oupu gap, and herefore achieve a zero welfare loss. This was because sabilizing prices led o a sable oupu gap, and vice-versa. Here, i is in general no possible o simulaneously sabilize price inflaion, he oupu gap, and wage inflaion. This is easy o see. For he oupu gap o be zero, he real wage mus equal is naural rae which is in urn equal o Ã. Bu for he real wage o equal is naural rae, eiher wages or prices mus adjus o he exen o which à moves around. Hence, you can simulaneously ge he real wage o flucuae which i mus if here are real shocks wihou eiher prices or wages moving around. In oher words, he presence of wage sickiness makes a cenral bank face a non-rivial radeoff wihou having o resor o including a cos-push shock in he model. Below I presen welfare losses from a quaniaive version of he model using he parameers described above, along wih ρ a =.95 and s a =., for differen ypes of moneary policy: Policy Taylor Rule -.2 Price Inflaion Targeing -.2 Wage Inflaion Targeing -. Gap Targeing -. For he Taylor rule specificaion I use ĩ = ρ i ĩ + ρ i φ π π + φ x X. This is prey ineresing in ha we see ha price inflaion argeing does very poorly. The reason why is fairly ransparen. If you arge zero price inflaion, hen he real wage gap mus be equal o zero from he price Phillips Curve. Bu a zero real wage gap means ha real wages mus move around onefor-one wih Ã. Given ha prices can move, his means we have o have a lo of wage inflaion, and he relaive weigh on wage inflaion is very high. Hence, price inflaion argeing does poorly. Wage inflaion argeing does very well, which makes sense given he high weigh on wage inflaion. Ineresingly, oupu gap argeing does well oo. Sabilizing he gap resuls in lile wage inflaion and comparaively much price inflaion, bu given he relaive weighs his ends up doing well from a welfare perspecive. As in he simpler model wih jus sicky prices, we can derive formal firs order condiions o characerize he opimal policy, bu I leave ha as an exercise for he ineresed reader. L 35