A New-Keynesian Model

Deparmen of Economics Universiy of Minnesoa Macroeconomic Theory Varadarajan V. Chari Spring 215 A New-Keynesian Model Prepared by Keyvan Eslami

A New-Keynesian Model You were inroduced o a monopolisic compeiion model wih price rigidiies in class where only a consan fracion of firms, he so-called flexible-price firms, could change heir prices afer he realizaion of a shock; he res had o se heir prices a he beginning of each period and keep heir promise of delivering he demand afer he realizaion of exogenous shocks. Today, we are going o presen a similar basic New-Keynesian model in which monopolisic firms can adjus heir prices in each period wih a consan probabiliy of 1 θ. This is based on he saggered price-seing model of Calvo 1983). Thus, afer he price is se, he expeced duraion of his price being effecive is 1/ 1 θ). This creaes a so-called!!) micro-founded rigidiy in aggregae price level. To make his ineresing, imagine here is a fairy in his economy, which we refer o as Calvo fairy! This ficiious being appears a he beginning of each period, chooses fracion 1 θ) of firms randomly and aps hem on he shoulder, giving hem he permission o change heir prices. This sory occurs in every period. A more realisic sory is ha here are small coss of changing prices in his economy, referred o as menu coss. So ha, in each period, only a fracion θ of firms find i oo cosly o adjus heir prices o unanicipaed shocks in economy. We will invie you o hink abou he difference of hese wo seings, in erms of persisence of unanicipaed nominal shocks in he economy! For a firm ha ses is price in period, here is a 1 θ) chance of reseing price nex period. This means here is a 1 θ) probabiliy of he price in being in effec only in period, wihou experiencing any rigidiy in price. Wih probabiliy θ, he firm has o keep is price fixed in period + 1. So, he chance of a one-period price rigidiy is θ 1 θ). Therefore, he expeced lengh of price rigidiy is θ k 1 θ) k, which is a geomeric power series converging o 1 θ) 1. We are going o sudy he implicaions of such assumpions for he neuraliy of moneary policy and nominal shocks in such an economy. 1

Households The economy is inhabied by an infiniely-lived represenaive household whose preferences over sreams of consumpion and labor are represened by he following uiliy funcion wih expeced form: E = where C is a consumpion index, given by [ C := β U C, N ), C i) 1 ] 1 di. Here, C i) is he consumpion of a differeniaed good indexed by i. We will assume here is an exogenous coninuum of such goods in he economy, of uni mass. The budge consrain of he household is given by P i) C i) di + Q B B 1 + W N + T, where p i) is he price of variey i, B is a one-period risk free bond wih face value price Q, W is he wage rae, and T is he lump-sum ransfers o he households possibly in erms of profis, ec.). Thus, we may wrie he problem of he household as: max E β U C, N ) = [ s.. C = C i) 1 ] 1 di P i) C i) di + Q B B 1 + W N + T lim E B ), T where he las condiion rules ou he possibiliy of Ponzi schemes in he economy. 2

Exercise 1 Wha is he difference beween condiion lim T E B ), and he no-ponzi scheme condiion you have seen previously e.g. in Tim s course)? Which one is more sric? We can hink of a household s problem in period as consising of wo pars: deciding abou he fracion of wealh household wans o allocae o consumpion, and dividing his fracion beween differen varieies of consumpion goods. We begin wih he laer; i.e. given amoun W of wealh dedicaed o consumpion and price of each differeniaed good, how would household divide W beween varieies. Given a consan amoun of wealh, W, spen on consumpion in period, P i) C i) di = W, we can wrie he problem of he household as choosing beween he differen varieies as: max Ci) U C, N) [ s.. C = C i) 1 ] 1 di P i) C i) di = W. The firs order condiion for he opimal consumpion of variey i is: U C C, N) C i) 1 [ C k) 1 ] +1 1 dk = λp i), i [, 1], where λ is he Lagrange muliplier on he consrain of he problem. By wriing he same condiion for variey j, and dividing he wo equaions, we have: C i) 1/ P i) = 1/ C j) P j), [ ] P j) C i) = C j). P i) 3

If we subsiue his resul ino he definiion of consumpion index, we ge: and, hence: where: C = C = [ [ [[ ] 1 ] 1 P j) C j)] di, P i) [ ] 1 P j) C j) 1 di P i) ] 1 P j) C j) = [ ] C, 1 P i)1 1 di [ ] P j) C j) = C, 1) P [ P :=, ] 1 1 P i) 1 di. 2) We will refer o Equaion 1) as he demand funcion for variey j, as a funcion of is price P j), aggregae consumpion index C, and aggregae price index P. To see he inuiion behind he aggregae price index, le us subsiue his resul back ino he consrain of he opimizaion problem above, o ge: W = = P i) C i) di [ ] P i) P i) Cdi P = P i)1 di [ P i)1 di [ = P i) 1 di =P.C. ] 1 C ] 1+ 1 C 4

Therefore, we may wrie he firs par of a household s problem simply as: max E β U C, N ) s.. = P C + Q B B 1 + W N + T lim E B ), T where P is defined in 2). The firs order condiions for his problem are: 1. β U C C, N ) = P λ, 2. β U N C, N ) = W λ, 3. and Q λ = s +1 λ +1 s +1 ) where s +1 is an index for nex period s sae). If we combine condiions 1 and 2, we ge he inraemporal opimaliy condiion for labor supply: U N C, N ) U C C, N ) = W P. 3) Moreover, we can derive he Euler equaion by combining 2 and 3: Q = βe UC C +1, N +1 ) U C C, N ) P P +1 ). 4) Remark To see how we can derive his, assume for he momen ha nex period s sae is given by s. Then: β U C C, N ) Q = β +1 U C C +1 s +1 ), N +1 s +1 )) P P s +1 s +1 ) +1 4) can be derived using his equaliy. =β +1 E U C C +1, N +1 ) P +1. Now, if we assume ha he period uiliy akes he following separable form, U C, N) = C1 σ 1 σ) N 1+φ 1 + φ), 5

hen, we can wrie 4) as: 1 = E β 1 Q C σ +1 C σ which can be wrien in naural log erms as: P P +1 ), 1 = E exp log β + i σ c +1 π +1 )). 5) The lower case leers indicae he logarihm of he variables. i is defined as log 1/Q ). Remark i is he ne nominal rae of reurn. To see why, noe ha he gross nominal rae of reurn is 1/Q. If we denoe his by 1 + i, hen: ) 1 log 1 + i ) = log Q i. On a balanced growh pah, where consumpion grows a a consan rae γ, we have: i = log β + σγ + π. A firs order Taylor expansion of 5) around he balanced growh pah yields: exp log β + i σ c +1 π +1 ) 1 + log β + i σ c +1 π +1. If we subsiue his resul ino Equaion 5), we ge a firs order log-linearizaion of his condiion: log β + i σe c +1 c ) E π +1 ) =, c = E c +1 ) 1 σ [log β + i E π +1 )]. 6) On he oher hand, under he assumpion of separable uiliy form, we may wrie Equaion 3) as: N φ C σ = W P. 6

In logarihmic erms, we can rewrie his as: w p = σc + φn. 7) To inroduce money explicily in he model, we add an ad-hoc log-linear money demand equaion o he household side of he economy as well: m p = y ηi. This equaion, deermines he pah of nominal ineres rae i as a funcion of exogenous money supply M in he equilibrium. To see he inuiion behind his money demand funcion, consider he following problem faced by a household ha values money holdings we have seen how such a uiliy form may emerge from a moneary economy): max s.. [ E β = C 1 σ 1 σ) + M /P ) 1 ν 1 ν) ] N 1+φ 1 + φ) P C + Q B + M B 1 + M 1 + W N + T lim E B ). T The firs order condiions for he household are: 1. β C σ = P λ, 2. β 1 P M P ) ν = λ λ +1, 3. and Q λ = λ +1. If we subsiue from 3 and 1 ino 2, we ge: β 1 ) ν M = 1 Q ) β 1 P P M P ) ν = 1 Q ) C σ. P C σ, 7

Using our definiion of nominal rae of reurn, we may wrie his condiion as: M /P ) ν C σ = 1 e i i. Now, if we ake he naural logarihm, we may wrie his equaion as follows: m p = σ ν c 1 ν log 1 e i) σ ν c 1 ν e i 1) i. Assuming uni elasiciy of real) money demand wih respec o consumpion i.e. σ/ν = 1) and incorporaing marke clearing condiion, hen, imply: m p = y ηi. Firms There is a coninuum of monopolisically compeiive firms in he economy ha produce a differeniaed good according o he same echnology, given by: Y i) = A N i). Firm i faces a demand funcion of he form in 1), and akes he aggregae price index P and aggregae consumpion index C as given. In each period, fracion 1 θ of he firms have he chance o adjus heir prices while fracion θ has o keep heir price fixed and mee he demand in he marke). Therefore, he average inerval beween price changes for a firm is 1/ 1 θ). In his sense, θ is an index of price sickiness in his economy. 8

The problem faced by a firm ha is chosen in period o adjus is price can be wrien as: max p s.. Y +k = θ k E Q,+k P Y +k Ψ +k Y+k ))) ) P C +k, P +k where Ψ is he cos funcion. Q,+k is he sochasic discoun facor; i deermines how fuure nominal profis are discouned in dae s uiliy erms, and can be evaluaed as: k 1 ) Q,+k =E Q +s s= k 1 UC C +s+1, N +s+1 ) =E βe +s U s= C C +s, N +s ) ) =β k UC C +k, N +k ) P E, U C C, N ) P +k P +s P +s+1 where he second equaliy follows from he Law of Ieraed Expecaions. I is clear ha any firm apped by he Calvo fairy o adjus is price would face he same problem in period. Therefore, all he firms adjusing prices in period would choose If we subsiue for Y +k ino he objecive funcion, we can rewrie he problem of he monopolis as: ) ) max P θ k E Q,+k P ) 1 P +kc +k Ψ +k P ) P +kc +k ))) The firs order condiion for his problem, wih respec o he conrol P yields:: ))) θ k E Q,+k 1 ) P ) P+kC +k + P ) 1 P+kC Ψ +k Y+k +k =. Y +k If we subsiue back Y +k, we ge: ))) θ k E Q,+k 1 ) Y +k + P ) 1 Ψ +k Y+k Y +k =. Y +k 9

If we muliply boh sides of he equaliy by P / 1 ), we have: )) θ k E Q,+k Y +k P 1) ψ +k =, where ψ +k := Ψ +k Y+k ) / Y+k is he nominal marginal cos of he firm in period + k. Noe ha, when θ =, his condiion gives he familiar fricionless opimal pricing condiion of P = ) Ψ +k Y+k ; 1) Y +k i.e. price of he monopolisic firm is marked up above he marginal cos, by a consan ha depends on he elasiciy of demand. We denoe his fricion-less mark-up by M. If we divide his equaion be P 1, we ge: P θ k E Q,+k Y +k M ψ )) +k P +k =. P 1 P +k P 1 By leing P +k /P 1 := Π 1,+k be he inflaion beween periods 1 and + k, we can rewrie his condiion as: )) P θ k E Q,+k Y +k M MC +k Π 1,+k =, 8) P 1 where MC +k is he real marginal cos in period + k of a firm ha ses is price a. Consider a seady-sae of his economy in which he inflaion rae is zero, Π 1,+k = 1; he price level remains consan, and, hence, P = P = P 1. Moreover, C = Y = Y +k remains a consan, and Q,+k = β k. In addiion, from firm s firs order condiion. Therefore: P P 1 M MC +k Π 1,+k =, MC +k = 1 M, 1

so ha, in he zero-inflaion seady-sae, he real marginal coss of all firms are equal and consan. We can log-linearize 8) around his zero-inflaion seady-sae as: p p 1 = 1 βθ) βθ) k ) E mc+k + p +k p 1, 9) where mc +k :=mc +k mc =mc +k + log M, is he log deviaion of marginal cos in + k from is seady-sae value, mc. Remark To log-linearize an equaion of he form f X, Y ) = g Z ) around he seadysae, use he following formula: [f 1 X, Y ) Xx + f 2 X, Y ) Y y ] g Z) Zz, where X,Y, and Z are he seady-sae values of X, Y, and Z, and small leers denoe he naural logarihms. Aggregae Price Level Dynamics Le S [, 1] be he firms ha are no adjusing heir prices in period. The aggregae price index in period, as we defined i before, is hen given by: P 1 = P i) 1 di = P 1 i) 1 di + S S c P i) 1 di. 11

Noing ha each firm ha is allowed o adjus is price would choose P in, and he measure of firms ha can adjus is 1 θ, we can wrie: S c P i) 1 di = P ) 1 S c di = 1 θ) P ) 1. On he oher hand: S P 1 i) 1 di =θ =θp 1 1. P 1 i) 1 di To see why his is he case, consider he firms all wih heir prices equal o P 1 ; suppose hese firms make a se od measure di. Of his se, exacly fracion θ of firms canno adjus heir price in. Therefore, a se of measure θdi of firms would have prices equal o P 1 in. These firms consiue fracion θdi/µ S ) of he se S where µ is he measure of se S ). If we repea his argumen for all he prices ha prevail in 1, we end up covering he enire se S. Thus: P 1 i) 1 di = S Hence, he aggregae price level in can be wrien as: P 1 i) 1 θdi. P 1 = θp 1 1 + 1 θ) P ) 1. By dividing his equaion be P 1, we ge: P P 1 ) 1 = θ + 1 θ) P P 1 ) 1, ) P Π 1 1 = θ + 1 θ). P 1 In he zero-inflaion seady sae, where Π = 1, P = P = P 1. So, if we log-linearize 12

around he seady-sae, we ge: 1 θ) 1 ) P P 1 1 P.p 1 θ) 1 ) P ) 1 P 1 2 P 1.p 1 = 1 ) Π Π.π, 1 θ) p p 1 ) = π. 1) Flexible Prices Consider his economy under he assumpion ha here are no nominal rigidiies. When he prices are compleely flexible i.e. θ = ), no dynamic decisions would be made, and we have: Y n i) = Y n = A N n, where n superscrips denoe he flexible or naural values of he variables, henceforh. Then, as noed before, all monopolisic firms would choose he same price in each period, which is a fixed mark-up above he marginal cos: P = Marginal cos is given by W /A. Thus: 1) P = Ψ Y n ) Y n W. 1) A. If we normalize W = 1, we ge: P = 1. 1) A Under he assumpion of a separable uiliy funcion, we saw ha he inraemporal decision for labor supply mus follow he following opimaliy condiion: N n ) φ 1) A N n σ = A, ) N n ) φ+σ 1) = A 1 σ. 13

If we subsiue from firms echnology for N n, we ge: Y n A 1) ) φ+σ = 1) A 1 σ, = Y n ) φ+σ A 1 φ, which, as we expec, is a consan value, independen of. Moreover, recall ha MC n = M was defined o be equal o he fricion-less mark-up equal o 1) /). Taking he logarihm of his equaion, we ge he real marginal cos under flexible prices: mc n = φ + σ) y n 1 + φ) a. On he oher hand, he real rae of reurn in his case is equal o E P / Q P +1 )), which can be calculaed as: ) R n P =E Q P +1 ) P 1 C n σ ) =E +1 P +1 P +1 β C n P = 1 ) C n σ ) β E +1. C n In he equilibrium, when C n as: = Y n, afer aking he logarihm, we can rewrie his equaion r n ) = log β + σe y n +1 y n. Equilibrium In equilibrium, he real marginal cos in period + k of a firm ha ses is price in is MC +k = W +k A +k P +k, 14

which is he same for all firms no maer a which dae hey are adjusing heir price), and, hence, equal o MC +k. In logarihms: mc +k = w +k p +k a +k = mc +k. If we subsiue his ino Equaion 9), we have: p p 1 = 1 βθ) βθ) k E mc +k ) + 1 βθ) βθ) k E p +k p 1 ). Noe ha, we can expand his equaion as: p p 1 = 1 βθ) βθ) k E mc +k ) + 1 βθ) βθ) k E p +k p +k 1 + p +k 1 p +k 2 +... + p p 1 ) = 1 βθ) βθ) k E mc +k ) + 1 βθ) βθ) k E π +k + π +k 1 +... + π ). Thus: p p 1 = 1 βθ) βθ) k E mc +k ) + 1 βθ) π + βθπ + β 2 θ 2 π +... ) + 1 βθ) βθπ +1 + β 2 θ 2 π +1 +... ) +... = 1 βθ) βθ) k E mc +k ) π + 1 βθ) 1 βθ) + βθ 1 βθ) π +1 1 βθ) +... = 1 βθ) βθ) k E mc +k ) + βθ) k E π +k ). 11) If we wrie his equaion for period + 1, and ake he expecaions wih respec o he 15

informaion available a, by Law of Ieraed Expecaions, we have: ) E p +1 p = 1 βθ) βθ) k+1 E mc +k+1 ) + βθ) k+1 E π +k+1 ), ) E p +1 p = 1 βθ) βθ) k E mc +k ) + Thus, we may wrie 11) as: k=1 k=1 βθ) k E π +k ). p p 1 = βθe p +1 p ) + 1 βθ) mc + π Now, if we subsiue from 1), we ge: ) π 1 θ) = βθe π+1 + 1 βθ) mc + π, 1 θ) 1 θ) 1 βθ) π = βe π +1 ) + mc. 12) θ If we solve his equaion forward, we ge a formula for inflaion as a discouned sum of expeced real mark-up deviaions: π = 1 θ) 1 βθ) θ β k E mc +k ). In he labor marke, in equilibrium, noing ha [ ] P i) Y i) = Y, P 16

we have: N = = Y A Y i) di [ ] P i) di. A If we ake logarihm of his equaion, we have: P n = y a + d, where d is a measure of price dispersion across firms. We can show ha, around he zeroinflaion seady-sae, d is zero up o a firs order approximaion. Thus, we may wrie his equaion as: n = y a. 13) If we subsiue from good s marke clearing condiion ino he household s opimaliy condiion for labor supply, we ge: w p = σy + φn. Afer subsiuion for n from Equaion 13), we ge: and, hence: w p =σy + φ y a ) = σ + φ) y φa, mc =w p a =σy + φ y a ) a = σ + φ) y 1 + φ) a. Noing ha, in he zero-inflaion seady-sae, he logarihm of real marginal cos is equal 17

o he real marginal cos under flexible prices, we can wrie: mc =mc mc =mc mc n = σ + φ) y y n ). We define he log-deviaion of oupu from is flexible price counerpar as oupu gap, and denoe i by ỹ := y y n. Thus: mc = σ + φ) ỹ. We may now wrie Equaion 12) as: π = βe π +1 ) + κỹ, 14) where κ := 1 θ) 1 βθ) θ This is he well-known New-Keynesian Phillips curve. σ + φ). On he oher hand, we can wrie Equaion 6), in equilibrium as: y = E y +1 ) 1 σ [log β + i E π +1 )]. Subracing y n from boh sides, we can wrie his equaion as: y y n = E y +1 ) E y n +1 ) + E y n +1 ) y n 1 σ [log β + i E π +1 )], ỹ = E ỹ +1 ) + E y n +1 ) 1 σ [log β + i E π +1 )], ỹ = E ỹ +1 ) 1 σ [i E π +1 ) r n ], 15) where r n := log β + σe y n +1 ), is he naural rae of ineres, as defined before. Equaion 15) is referred o as he dynamic 18

IS equaion. If we assume ha he effec of nominal rigidiies vanishes asympoically, lim T E ỹ +T ) =, we can solve his equaion forward o ge: ỹ = 1 σ r+k r+k n ), where r := i E π +1 ), is he real ineres rae his is he celebraed Fisher equaion); i.e. he oupu gap is proporional o he sum of curren and anicipaed deviaions beween he real rae of reurn and is counerpar under flexible prices. Equaions 14) and 15) are he building blocks of his basic New-Keynesian model and, as i urns ou, majoriy of New-Keynesian models); he New-Keynesian Phillips curve and dynamic IS equaion, ogeher wih a policy equaion ha deermines he dynamics of nominal ineres rae i, characerize he equilibrium in his economy. Given he exogenous process governing A, we know how y n evolves over ime. This, in urn, deermines he process governing he naural rae of reurn. Then, Equaions 14) and 15), ogeher, deermine how he inflaion and oupu gap would change hrough ime. The process governing i, in urn, depends on how moneary policy is conduced, hrough he ad hoc money demand equaion inroduced in he household s secion); his is in conras o classical models where moneary policy is neural. Here, moneary policy is a key deerminan of real changes in he economy. For insance, an equaion usually noed in he lieraure describing how moneary auhoriy reacs o he inflaion rae and oupu gap is he following: i = log β + α π π + α y ỹ + ν, where α π and α y are non-negaive coefficiens, and v is an exogenous componen wih zero mean. This is he well-known so-called Taylor rule we discussed why i is no appealing o call his a rule in class). 19

References Calvo, Guillermo A., Saggered Prices in a Uiliy-Maximizing Framework, Journal of Moneary Economics, 1983, 12 3), 383 398. 2