The Inflation Bias under Calvo and Rotemberg Pricing

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1 The Inflaion Bias under Calvo and Roemberg Pricing Campbell Leih Ding Liu June 19, 014 Absrac New Keynesian models rely heavily on wo workhorse models of nominal ineria - price conracs of random duraion (Calvo, 1983) and price adjusmen coss (Roemberg, 198) - o generae a meaningful role for moneary policy. These alernaive descripions of price sickiness are ofen used inerchangeably since, o a firs order of approximaion hey imply an isomorhpic Phillips curve and, if he seady-sae is efficien, idenical objecives for he policy maker and as a resul in an LQ framework, he same policy conclusions. In his paper we compue ime-consisen opimal moneary policy in benchmark New Keynesian models conaining each form of price sickiness. Using global soluion echniques we find ha he inflaion bias problem under Calvo conracs is significanly greaer han under Roemberg pricing, despie he fac ha he former ypically exhibis far greaer welfare coss of inflaion. The raes of inflaion observed under his policy are non-rivial and sugges ha he model can comforably generae he raes of inflaion a which he problemaic issues highlighed in he rend inflaion lieraure emerge, as well as he movemens in rend inflaion emphasized in empirical sudies of he evoluion of inflaion. Finally, we consider he response o cos push shocks across boh models and find hese can also be significanly differen. The choice of which form of nominal ineria o adop is no innocuous. Key words: New Keynesian Model; Moneary Policy; Roemberg Pricing; Calvo Pricing; Inflaion Bias; Time-Consisen Policy. JEL codes: E5, E63 We are graeful for commens from Guido Ascari, Fabrice Collard, Richard Dennis, Charles Nolan and seminar paricipans a he Universiy of Glasgow. Address: Economics, Adam Smih Business School, Wes Quadrangle, Gilber Sco Building, Universiy of Glasgow, Glasgow G1 8QQ. campbell.leih@glasgow.ac.uk. Address: Economics, Adam Smih Business School, 510, Roberson Building, Universiy of Glasgow, Glasgow G11 6AQ. dingliumah@gmail.com 1

2 1 Inroducion Mainsream macroeconomic analysis of boh moneary and fiscal policy relies heavily on he New Keynesian model. The disinguishing feaure of his model, relaive o a more classical approach, is ha i conains some form of nominal ineria. This allows moneary policy o have real effecs, and widens he degree of ineracion beween moneary and fiscal policies, since moneary policy affecs boh he size of he ax base and real deb service coss in such models. Typically, one of wo workhorse forms of nominal ineria are adoped in he lieraure - Calvo (1983) price conracs, and Roemberg (198) price adjusmen coss. In he former, firms are only able o adjus heir prices afer random inervals of ime, such ha, ouside of a zero inflaion seady-sae here will be a cosly dispersion of prices across firms. While he laer implies ha all firms behave symmerically in seing he same price, bu ha hey face quadraic adjusmen coss in doing so. Despie his fundamenal difference, researchers have ypically reaed he wo approaches as being equivalen since he New Keynesian Phillips Curve (NKPC) hey imply are, o a firs order of approximaion, isomorphic when linearized around a zero inflaion seady sae. Moreover, when ha zero inflaion seady-sae is also efficien (ha is, i maches he oupu level ha would be chosen by a benevolen social planner) i can be shown ha he second order approximaion o welfare rewrien in erms of inflaion and he oupu gap is also he same across he wo approaches (see Nisico, 007). Under hese condiions, o a firs order of approximaion, he wo approaches would yield he same policy implicaions. For hese reasons he wo approaches have largely been reaed as synonymous wihin he New Keynesian lieraure. However, despie his broad consensus, here are examples wihin he lieraure where he wo approaches do differ. The firs is where he seady-sae around which we approximae he New Keynesian economy is no efficien. For example, Lombardo and Vesin (008) relax he assumpion of Nisico (007) and consider he second order approximaion o welfare when he seady sae is no efficien. They find ha he coss of such inefficiencies are ypically larger in he Calvo economy. This mirrors he resuls in Damjanovic and Nolan (011). The second assumpion underpinning he equivalence resul, is ha he economy is approximaed around a zero inflaion seady sae (or ha any seady-sae inflaion is perfecly indexed and herefore cosless, see Yun (1996)). A lieraure considering he imporance of rend inflaion argues ha his is no he case, and ha he implicaions of failing o accoun for rend inflaion can be dramaic, see Ascari and Sbordone (013) for a survey. The presence of even a modes degree of (unindexed) seady-sae inflaion can radically overurn deerminacy resuls, undermine he learnabiliy of raional expecaions equilibria, affec he moneary policy ransmission mechanism and change he naure of opimal policy. Moreover, hese effecs can differ across he wo forms of nominal ineria (Ascari and Rossi, 01) wih he larger impac of rend inflaion being fel under Calvo. The large coss of rend inflaion under Calvo is also refleced in he analysis of Damjanovic and Nolan (010b) where he seigniorage maximizing rae of inflaion is a double digi levels under Roemberg pricing, bu only single digis under Calvo. In shor here appears o be significan non-lineariies in he New Keynesian model which are affeced by he size of he seady-sae disorion, he degree of unindexed inflaion and he ype of nominal ineria adoped. However, his evidence largely comes from sudies which linearize such economies, eiher o a firs or second order approximaion, afer allowing for such facors.

3 In his paper we solve he benchmark New Keynesian model non-linearly using he wo sandard approaches o modelling price sickiness. Since we are no imposing any kind of approximaion around a seady-sae we can see clearly he exen o which he wo approaches differ. Moreover, raher han consider he Ramsey problem or commimen o a simple moneary rule, we shall consider ime-consisen opimal policy (commonly known as discreion). This in urn, given ha we are no using any arificial devices o ensure he model s seady-sae is efficien, implies ha we can measure he exen of he inflaionary bias problem under he wo forms of nominal ineria. This idenifies he exen o which a policy maker who is consrained o be ime-consisen would be unable o preven a cosly rise in seady-sae or rend inflaion. This is an imporan measure of he non-lineariies across he wo descripions of pricing behavior, bu also serves as a plausibiliy check on he relevance of he effecs highlighed in he lieraure on rend inflaion. The inflaion bias hus measures he maximum level of unindexed inflaion ha a policy maker would be forced o olerae - he policy maker which allowed inflaion o rise above his level is behaving sub-opimally even given he consrain ha hey canno commi. Therefore if he level of inflaion bias is significanly below ha required o generae he perverse resuls found in he rend-inflaion lieraure hen we would need o find a reason why policy makers are no only failing o commi, bu are generaing inflaion levels well beyond he maximum inflaion bias before we need worry abou hese properies of he New Keynesian model. While if he model implies a sizeable inflaion bias hen he issues raised by he rend inflaion lieraure and, more generally, he nonlineariies inheren in he New Keynesian model need o be aken more seriously. There is also an empirical lieraure which focusses on hese wo disorions in helping o explain inflaion dynamics. Ireland (007) allows for ime variaion in he Fed s inflaion arge o explain he evoluion of US inflaion. Cogley and Sargen (00) argue ha much of he movemen in US inflaion reflecs movemens in an underlying rend, raher han in flucuaions relaive o ha rend. While several auhors have sough o idenify he level of rend inflaion using generalizaions of he new Keynesian Phillips curve which allow for ime varying (unindexed) rend inflaion. As an example of he findings of his lieraure, Cogley and Sbordone (008) argue ha rend inflaion raher han any kind of backward-looking indexaion behavior is a major componen of observed movemens in inflaion. Again we can ask - can he benchmark model, using eiher Roemberg or Calvo pricing plausibly deliver he size of unindexed seady sae or rend inflaion hese papers infer o explain he daa? Moving away from he Ramsey descripion of policy is imporan as such a policy implies ha he opimal rae of inflaion he policy maker would commi o would be zero in he benchmark model employing eiher Calvo or Roemberg pricing (Woodford, 003) and in he case of Calvo conracs very close o zero in models wih oher disorions due o, for example, fiscal policy (Schmi-Grohe and Uribe, 004) or a desire o generae seigniorage revenues (Damjanovic and Nolan, 011). Under Roemberg, he example of Damjanovic and Nolan suggess ha since he welfare coss of nominal ineria do no rise as sharply as he rae of inflaion rises under Roemberg, ha his may no be a general resul across he wo descripions of nominal ineria. Neverheless, he fac remains ha Ramsey policy would ypically imply ha inflaion was far lower and sable han appears o be found in he daa. 1 1 Chen e al. (014) assess he relaive empirical performance of a New Keynesian model wih habis and inflaion ineria wih policy described by no only by simple rules, bu also opimal policy under discreion, commimen and quasi-commimen. They find ha discreion fis he daa far beer han 3

4 There are some recen papers using global soluion echniques which also consider opimal discreionary policy in he benchmark model under Calvo conracs - see Van Zandweghe and Wolman (011) and Anderson e al. (010), which is hen exended in Ngo (014) o allow for discoun facor shocks which imply ha policy mus accoun for he zero lower bound (ZLB). Oher auhors also consider issues relaing o he ZLB in models which use Roemberg pricing, bu also inroduce exensions such as capial (see Gavin e al. (013), Braun and Korber (011), Johannsen (014)), consumpion habis (Gus e al. (01) and Aruoba and Schorfheide (013)), labor marke fricions (Roulleau-Pasdeloup (013)) or fiscal policy (Nakaa (013), Niemann e al. (013) and Johannsen (014)). 3 Solving non-linear represenaions of an enriched New Keynesian model is ypically far more compuaionally inensive han convenional perurbaion mehods, and hese laer auhors have all adoped he Roemberg descripion of price sickiness since his reduces he number of sae variables one mus consider. Furhermore, in calibraing he Roemberg price adjusmen cos parameer almos all hese auhors use a convenional parameerizaion which maches he slope of he linearized NKPC across he Roemberg and Calvo varians of he New Keynesian model afer assuming a zero inflaion seadysae. In oher words he lieraure is ypically implicily assuming ha he equivalence of he wo forms of nominal ineria is reained in non-linear soluions of he New Keynesian model where he seady-sae is disored and he rae of inflaion will ypically no be zero. To our knowledge, he curren paper is he firs o formally compare and conras ime-consisen opimal policy under he wo forms of price-seing using global soluion algorihms and herefore o assess how innocuous he choice of one form of price-seing over he oher acually is. We find ha he inflaionary bias problem is non-rivial under boh descripions of nominal ineria, bu is much greaer under Calvo. This is despie earlier resuls implying ha he coss of inflaion are much higher under Calvo han Roemberg. This essenially arises because of he differen average mark-up behavior under he wo models. Under Calvo higher inflaion causes hose firms who are able o adjus prices in a paricular period o raise ha price in anicipaion of no being able o readjus ha price for a prolonged period despie he general rise in he price level. This leads o an increase in he average mark-up as inflaion rises. In conras, under Roemberg all firms se he same price, period by period, bu face adjusmen coss in doing so. In discouning fuure profis hey also discoun fuure price adjusmen coss. As a resul in he face of higher inflaion he firms pospone some of he required price adjusmen due o his discouning effec, which serves o reduce he average markup. Accordingly, for a given degree of monopolisic compeiion which induces an inflaion bias, his furher raises (lowers) he markup under Calvo (Roemberg) and hereby worsens (improves) he inflaionary bias problem. This effec also ends o imply ha he inflaionary impac of a given cos-push shock is greaer under Calvo pricing, ceeris paribus. While he presence of an addiional sae variable under Calvo price-seing, namely price dispersion, can also resul in a hump-shaped response in oupu o cos-push shocks which would no be he any oher descripion of policy, especially commimen which is simply oo effecive in sabilising he economy o be consisen wih he daa. Fernandez-Villaverde e al. (01), Wieland (013) and Richer e al. (013) also explore equilibrium dynamics around he ZLB in varians of he New Keynesian model which adop Calvo price conracs, bu which adop a rule-based descripion of policy. 3 Wihin his group, Shibayama and Sunakawa (01), Nakaa (013), and Niemann e al. (013) explore opimal policy in various New Keynesian models using Roemberg pricing. The ohers uilise a rule-based descripion of policy. 4

5 case under eiher Roemberg pricing or he benchmark linearized model. The fac ha seady-sae inflaion would ceeris paribus, and using sandard calibraion approaches, be significanly higher under Calvo also has implicaions for he probabiliy of hiing he ZLB such ha sudies adoping Roemberg pricing are more likely o experience such episodes. The res of he paper is organized as follows. In secion, we describe he basic model under boh Calvo and Roemberg pricing. In secion 3, we formulae he opimal discreionary policy problem wih Roemberg and Calvo pricing, respecively. In secion 4, we presen numerical resuls. In secion 5, we exend he analysis o allow for a ax-driven cos-push shock o assess policy rade-offs. We conclude in secion 6. The Model This secion describes he basic economic srucure in our model..1 Households There are a coninuum of households of size one. We shall assume full asse markes, such ha, hrough risk sharing, hey will face he same budge consrain and make he same consumpion plans. As a resul, a period 0 he ypical household will seek o maximize he following objecive funcion, E 0 =0 β U(C, N ) (1) where 0 < β < 1 denoes he discoun facor, C and N are a consumpion aggregae, and labour supply a period, respecively. The household purchases differeniaed goods in a reail marke and combines hem ino composie goods using a CES aggregaor: ( 1 C = 0 ) ɛ C (j) ɛ 1 ɛ 1 ɛ dj, ɛ > 1 () where C (j) is he demand for differeniaed goods of ype j. The elasiciy of subsiuion beween varieies ɛ can be assumed o be ime varying if we wish o allow for cos-push or mark-up shocks, bu here we hold i fixed. The budge consrain a ime is given by 1 0 P (j)c (j)dj + E {Q,+1 D +1 } = Ξ + D + W N T (3) where P (j) is he nominal price of ype j goods, D +1 is he nominal payoff of he nominal bonds porfolio held a he end of period, Ξ is he represenaive household s share of profis in he imperfecly compeiive firms, W are wages, and T are lump-sum axes. Q,+1 is he sochasic discoun facor for one period ahead payoffs. The labor marke is perfecly compeiive and wages are fully flexible. Households mus firs decide how o allocae a given level of expendiure across he various goods ha are available. They do so by adjusing he share of a paricular good 5

6 in heir consumpion bundle o exploi any relaive price differences his minimizes he coss of consumpion. The demand curve for each good j is, where he aggregae price level P is defined o be ( ) ɛ P (j) C (j) = C (4) ( 1 P = 0 P ) 1 P (j) 1 ɛ 1 ɛ dj. (5) The dynamic budge consrain a period can herefore be rewrien as.1.1 Households problem P C + E {Q,+1 D +1 } = Ξ + D + W N T. (6) The household s decision problem can be deal wih in wo sages. Firs, regardless of he level of C he household purchases he combinaion of individual goods ha minimizes he cos of achieving his level of he composie good. Second, given he cos of achieving any given level of C, he household chooses C, D +1 and N opimally. We have solved he firs sage problem above. For racabiliy, we assume ha (1) akes he specific form E 0 =0 β ( C 1 σ 1 1 σ N 1+ϕ ). (7) 1 + ϕ where σ > 0 is a risk aversion parameer and ϕ > 0 is he inverse of he Frisch elasiciy of labor supply. We can hen maximize uiliy subjec o he budge consrain (6) o obain he opimal allocaion of consumpion across ime, ( ) σ ( ) C P β = Q,+1. C +1 P +1 Taking condiional expecaions on boh sides and rearranging gives {( ) σ ( )} C P βr E = 1, (8) C +1 P +1 1 where R E (Q,+1 is he gross nominal reurn on a riskless one period bond paying ) off a uni of currency in + 1. This is he familiar consumpion Euler equaion which implies ha consumers are aemping o smooh consumpion over ime such ha he marginal uiliy of consumpion is equal across periods (afer allowing for iling due o ineres raes differing from he households rae of ime preference). The second firs order condiion concerning labour supply decision is given by ( ) W = N ϕ C σ. (9) P 6

7 . Firms Each firm produces a differeniaed good j using a consan reurns o scale producion funcion: Y (j) = A N (j) (10) where Y (j) is he oupu of firm j, and N (j) denoes he hours hired by he firm, A is an exogenous aggregae produciviy shock a period, and a = log(a ) is ime varying and sochasic 4. Similar o he household s problem, we firs consider he cos minimizaion problem of firm j, which implies min {N (j)} ( W P ) N (j) s.. Y (j) A N (j). mc = W, (11) P A where mc is he Lagrange muliplier and also he real marginal cos of producion. Noe ha he real marginal cos described in (11) does no depend on he oupu level of an individual firm, so long as is producion funcion exhibis consan reurns o scale and prices of inpus (here labor) are fully flexible. The demand curve he firm j faces is given by ( ) ɛ P (j) Y (j) = Y, ( 1 where Y = Y 0 (j) ɛ 1 ɛ ) ɛ ɛ 1 dj. The inermediae-good secor is monopolisically compeiive and he inermediae good producer herefore has marke power. In he following, we consider wo alernaive forms of price sickiness - firsly ha due o Roemberg (198) and hen ha of Calvo (1983)...1 Roemberg Pricing The Roemberg model assumes ha a monopolisic firm faces a quadraic cos of adjusing nominal prices, which can be measured in erms of he final good and given by ( ) φ P (j) P 1 (j) 1 Y (1) where φ 0 measures he degree of nominal price rigidiy. The adjusmen cos, which accouns for he negaive effecs of price changes on he cusomer firm relaionship, increases in magniude wih he size of he price change and wih he overall scale of economic aciviy Y. The problem for firm j is hen o maximize he discouned value of nominal profis, max E {P (j)} =0 P s=0 Q,+s Ξ +s 4 Typically, he logarihm of A is assumed o follow an AR(1) process: a = ρ a a 1 + e a, 0 ρ a < 1 where echnology shock e a is an i.i.d. random variable, which has a zero mean and a finie sandard deviaion σ a. 7

8 where nominal profis are defined as Ξ = P (j)y (j) mc Y (j)p φ ( ) P (j) P 1 (j) 1 Y P (13) = P (j) 1 ɛ P ɛ Y mc P (j) ɛ P 1+ɛ Y φ ( ) P (j) P 1 (j) 1 Y P. Firms can change heir price in each period, subjec o he paymen of he adjusmen cos. Hence, all he firms face he same problem, and hus will choose he same price, and produce he same quaniy. In oher words, P (j) = P and Y (j) = Y for any j. Hence, he firs-order condiion for a symmeric equilibrium is ( ) σ ] C Y +1 (1 ɛ) + ɛmc φπ (Π 1) + φβe Π +1 (Π +1 1) = 0. (14) C +1 Y This is he Roemberg version of he non-linear Phillips curve ha relaes curren inflaion o fuure expeced inflaion and o he level of oupu... Calvo Pricing Each period, he firms ha adjus heir price are randomly seleced, and a fracion 1 θ of all firms adjus while he remaining θ fracion do no adjus. Those firms ha do adjus heir price a ime do so o maximize he expeced discouned value of curren and fuure profis. Profis a some fuure dae + s are affeced by he choice of price a ime only if he firm has no received anoher opporuniy o adjus beween and + s. The probabiliy of his is θ s. The firm s pricing decision problem hen involves picking P (j) o maximize discouned nominal profis Using he demand curve for he firm s produc, his objecive funcion can be wrien as ( ) ɛ ( ) ɛ θ s P (j) P (j) Q,+s P (j) Y +s mc +s Y +sp +s]. E s=0 P +s ( ) σ where he discoun facor Q,+s is given by β s C P C +s P +s, and mc +s is he marginal cos of producion. Le P be he opimal price chosen by all firms able o rese heir price a ime. The firs order condiion for he opimal choice of P is, where ( P ɛ = P ɛ 1 ) K p F p P +s ( ) ɛ ] K p = C σ P+1 mc Y + θβe K p +1 F p P (P+1 ) ɛ 1 = C σ Y + θβe F p +1 P ]. (15) 8

9 The price index evolves according o ( ) P 1 ɛ 1 = (1 θ) + θ(π ) ɛ 1 wih Π P P. P 1 (16) and price dispersion is described by 1 ( P (j) 0 P.3 Aggregae Condiions ) ɛ dj = (1 θ) ( P P ) ɛ ( ) ɛ P + θ 1. (17) P 1 Under Roemberg pricing, as all he firms will employ he same amoun of labour, he aggregae producion funcion is simply given by Y = A N. and he aggregae resource consrain is given by Y = C + φ (Π 1) Y. Noe ha he Roemberg adjusmen cos creaes an inefficiency wedge ψ R beween oupu and consumpion C = ( ) 1 ψ R Y = ( ) 1 ψ R A N (18) where ψ R = φ (Π 1). In he case of Calvo pricing, firms changing prices in differen periods will generally have differen prices. Thus, he model feaures price dispersion. When firms have differen relaive prices, here are disorions ha creae a wedge beween he aggregae oupu measured in erms of producion facor inpus and aggregae demand measured in erms of he composie goods. Specifically, N (j) = Y (j) A = ( ) ɛ P (j) Y which yields, 1 N = N (j)dj = Y 1 ( ) ɛ P (j) dj = Y 0 A 0 P A afer inegraing across firms. 1 implies ha price dispersion is always cosly in erms of aggregae oupu: he higher, he more labour is needed o produce a given level of oupu. Moreover, under Calvo differen firms wih differen prices will employ differen amouns of labor. This explains why higher price dispersion acs as a negaive produciviy shif in he aggregae producion funcion: Y = (A / )N. In addiion, price dispersion is a backward-looking variable, and inroduces an inerial componen ino he model. Under Calvo, he aggregae resource consrain is simply given by Y = C. Hence, define ψ c = 1 as an inefficiency wedge under Calvo, hen P A C = Y = A N (1 + ψ c ) (19) 9

10 Comparing (18) and (19), i is illuminaing o noe ha he Roemberg adjusmen cos creaes a wedge ψ R beween aggregae consumpion and aggregae oupu, while he Calvo price dispersion creaes a wedge ψ c beween aggregae hours and aggregae oupu. In addiion, boh wedges are non-linear funcions of inflaion, and hey are minimized a one when seady-sae inflaion equals zero (Π = 1), and boh wedges increase as rend inflaion moves away from zero. See Ascari and Rossi (01) for a discussion. Appendix C.1 summarizes he models under Roemberg and Calvo pricing. 3 Opimal Policy Problem Under Discreion Under discreion, he moneary auhoriy solves a sequenial or period-by-period opimizaion problem, which maximizes he represenaive household s expeced discouned uiliy subjec o he opimaliy condiions from marke paricipans, he aggregae condiions, and he law of moion for he sae variables. Therefore, under opimal discreion, he policymaker canno commi o a plan in he hope of influencing economic agens expecaions. 3.1 Roemberg Pricing Le V (A ) represens he value funcion a period in he Bellman equaion for he opimal policy problem. The opimal moneary policy hen solves he following opimizaion problem: { C 1 σ 1 V (A ) = max (Y } /A ) 1+ϕ + βe V (A +1 )] (0) {C,Y,Π } 1 σ 1 + ϕ subjec o, C = 1 φ (Π 1) ] Y (1) and, ( ) σ ] (1 ɛ) + ɛy ϕ C σ A ϕ 1 C Y +1 φπ (Π 1) + φβe Π +1 (Π +1 1) = 0 () C +1 Y Defining an auxilliary funcion, we can rewrie he Phillips curve () as, M(A +1 ) C σ +1Y +1 Π +1 (Π +1 1) (1 ɛ) + ɛy ϕ C σ A ϕ 1 φπ (Π 1) + φβc σ Y 1 E M(A +1 )] = 0 which capures he fac ha he policy maker recognizes ha any change in he sae variable will affec expecaions, bu canno promise o behave in a paricular way omorrow in order o influence expecaions oday. The opimal policy problem can hen be formulaed as he following Lagrangian, L = C1 σ 1 (Y { /A ) 1+ϕ + βe V (A +1 )] + λ 1 1 φ ] } 1 σ 1 + ϕ (Π 1) Y C { + λ (1 ɛ) + ϕ ɛy C σ A ϕ 1 φπ (Π 1) + φβc σ Y 1 E M(A +1 )] } 10

11 where λ 1 and λ are he Lagrange mulipliers. The firs order condiions and complemenary slackness condiions are given as follows, C σ = λ 1 λ { σɛy ϕ C σ 1 A ϕ 1 + σφβc σ 1 Y 1 E M(A +1 )] }, Y ϕ A 1 ϕ = λ 1 1 φ (Π 1) ] + λ { ɛϕy ϕ 1 C σ A ϕ 1 φβc σ Y E M(A +1 )] }, λ 1 φ (1 Π ) Y = λ φ (Π 1), C = 1 φ ] (Π 1) Y, 0 = (1 ɛ) + ɛy ϕ C σ A ϕ 1 φπ (Π 1) + φβc σ Y 1 E M(A +1 )]. Noe ha consumpion Euler equaion is non-binding from he poin of view of maximizing uiliy, because R (a variable of no direc ineres in uiliy) can effecively be chosen o achieve he desired level of consumpion. The fully nonlinear problem is hen o find five policy funcions which relae he hree choice variables {Y, C, Π } and wo Lagrange mulipliers {λ 1, λ } o he sae variable A, ha is, Y = Y (A ), C = C(A ), Π = Π(A ), λ 1 = λ 1 (A ), and λ = λ (A ). We will use he Chebyshev collocaion mehod o approximae hese five ime invarian rules. 3. Calvo Pricing Le V ( 1, A ) denoe he value funcion a period in he Bellman equaion for he opimal policy problem. The opimal moneary policy under discreion hen can be described as a se of decision rules for {C, Y, Π, P P, K p, F p, } which maximize, { C 1 σ 1 V ( 1, A ) = max ( } Y /A ) 1+ϕ + βe V (, A +1 )] 1 σ 1 + ϕ subjec o he following consrains, Resource consrain: Phillips curve: wih Y = C ( ) P ɛ K p = P ɛ 1 F p K p = ( Y ) ϕ A ϕ 1 Y + θβe M(, A +1 )] F p = Y C σ + θβe L(, A +1 )], where we have uilized wo auxilliary funcions, M(, A +1 ) = (Π +1 ) ɛ K p +1 and L(, A +1 ) = (Π +1 ) ɛ 1 F+1, p which highlighs he fac ha he policy maker recognizes ha any change in he sae variable will affec expecions, bu canno promise o behave in a paricular way omorrow in order o influence expecaions oday. Inflaion: ( ) P 1 ɛ 1 = (1 θ) + θ(π ) ɛ 1 P 11

12 Price dispersion: = (1 θ) ( ) P ɛ + θ (Π ) ɛ 1. P As before, he policy problem can be wrien in Lagrangian form as follows: L = C1 σ 1 ( Y /A ) 1+ϕ + βe V (, A +1 )] 1 σ 1 + ϕ + λ 1 Y C ] ( ) P + λ ɛ K p ] P ɛ 1 F p { + λ 3 K p ( Y ) ϕ A ϕ 1 Y θβe M(, A +1 )] } { + λ 4 F p Y C σ θβe L(, A +1 )] } ( ) ] P 1 ɛ + λ 5 1 (1 θ) θ(π ) ɛ 1 + λ 6 (1 θ) P ( P P ) ɛ θ (Π ) ɛ 1] where λ j (j = 1,.., 6) are he Lagrange mulipliers. The firs order condiions are given as follows: for consumpion, oupu, opimal price, inflaion, ( /A ) 1+ϕ Y ϕ C σ λ + (1 θ)(ɛ 1) numeraor of opimal price K p, λ 1 + σy C σ 1 λ 4 = 0 + λ 1 (1 + ϕ)( Y ) ϕ A ϕ 1 λ 3 C σ λ 4 = 0 ( P P ) ɛ ( ) P ɛ 1 λ 5 + ɛ(1 θ) λ 6 = 0 P (ɛ 1)θλ 5 ɛθ 1 Π λ 6 = 0 ( ) ɛ 1 ɛ 1 F p λ + λ 3 = 0 denominaor of opimal price F p, ( ) ɛ K p ɛ 1 (F p ) λ + λ 4 = 0 and price dispersion, 0 = (Y /A ) 1+ϕ ϕ + β E V (, A +1 )] ϕ( ) ϕ 1 A ϕ 1 Y ϕ+1 θβ E M(, A +1 )] λ 3 θβ E L(, A +1 )] λ 4 + λ 6 1 λ 3

13 Noe ha he envelope heorem yields V ( 1, A ) 1 = θ (Π ) ɛ λ 6 which allows us o rewrie he firs order condiion for price dispersion as, 0 = (Y /A ) 1+ϕ ( ) ϕ θβe (Π +1 ) ɛ λ 6+1 ] ϕ( ) ϕ 1 A ϕ 1 Y ϕ+1 λ 3 θβ E M(, A +1 )] λ 3 θβ E L(, A +1 )] λ 4 + λ 6 4 Numerical Analysis 4.1 Soluion Mehod We can solve he nonlinear sysem consising of hese seven firs order condiions and he six consrains o yield he ime-consisen opimal policy under Calvo pricing. Specifically, wihou commimen, we need o find hese hireen ime-invarian policy rules which are funcions of he wo sae variables { 1, A }. Tha is, we need o find policy funcions such as F P = F P ( 1, A ), K P = K P ( 1, A ), and Π = Π ( 1, A ). Similar o he Roemberg case, he Chebyshev collocaion mehod will be used o approximae hese policy funcions. We use he Chebyshev collocaion mehod o globally approximae he policy funcions. 5 In conras o he linear-quadraic approximaion mehod, his projecion mehod can capure he exen o which he wo approaches o modelling price sickiness differ, due o he non-lineariies inheren in he New Keynesian models. Firs, we discreize he sae space ino a se of collocaion nodes. In he Roemberg model, here is one sae variable (A ), while in he Calvo model here are wo sae variables ( 1, A ). Accordingly, he space of he approximaing funcions for he Roemberg pricing consiss of one-dimensional Chebyshev polynomials. In comparison, he space of approximaing funcions for he Calvo pricing is wo-dimensional, and is, given by he ensor producs of wo ses of Chebyshev polynomials. Then we define he residual funcions based on he equilibrium condiions. Gaussian-Hermie quadraure is used o approximae expecaion erms. Under Calvo pricing, he parial derivaives wih respec o price dispersion, are approximaed by differeniaing he Chebyshev polynomials. Finally, we solve he resulan sysem of nonlinear equaions consising of he residual funcions evaluaed over all he collocaion nodes 6. See appendix C. for deails. 4. Numerical Resuls 4..1 Benchmark Parameers and Soluion Accuracy The benchmark parameers for Calvo pricing are aken from Anderson e al. (010) and are sandard. We conduc a sensiiviy analysis below. To make he resuls from 5 Judd (199) and Judd (1998) are good references. 6 We also ried he ime ieraion mehod. Tha is, a smaller sysem of nonlinear equaions, composed of he residual funcions evaluaed a each collocaion node, is solved repeaedly. For he benchmark case in his paper, boh mehods find idenical soluions. However, he ime ieraion mehod will be used for oher cases since i is generally faser and more robus. 13

14 Roemberg pricing comparable, he value of price adjusmen cos is assigned so ha he linear quadraic approximaion for boh cases are equivalen 7. This implies an equivalence beween he wo forms of pricing provided he seady-sae is undisored wih a rae of inflaion of zero. Such an approach is ypically adoped in he lieraure even where auhors are considering models where hese condiions are no me. Table 1 summarizes he relevan parameer values. Wih his benchmark parameerizaion, we solve he fully nonlinear models via Chebyshev collocaion mehod. Following Anderson e al. (010), he relaive price dispersion is bounded by 1, 1.0], and he logged produciviy a akes values from σ a /(1 ρ a ), σ a /(1 ρ a )] = 0.4, 0.4]. For he Roemberg case, he order of approximaion n a is chosen o be 6, and he number of nodes for Gauss-Hermie quadraure q = 1. This combinaion is quie accurae, since he maximum Euler equaion error is on he order of For he Calvo case, he order of approximaion n a and n are boh assigned o be 6, and q = 1 for Gauss-Hermie quadraure. The maximum Euler equaion error over he full range is on he order of As suggesed by Judd (1998), his order of accuracy is reasonable. 4.. Seady Sae Inflaion Bias Figure 1 illusraes he soluion of he discreionary equilibria for he Calvo case. Similar o he resuls in Anderson e al. (010), he red doed line plos he value of as a funcion of 1 in a narrow inerval of 1, 1.0]. The seady sae relaive price dispersion is abou which is he inersecion poin beween he red line and he 45- degree solid line. A his fixed poin, he value of opimal gross inflaion Π (he dashed line) is abou , implying an annualized inflaion rae of.%. In conras, he discreionary inflaion rae for he Roemberg case is or 1.89% per year. I is well known ha he opimal rae of inflaion under commimen is zero, hence he inflaion bias is equal o he opimal rae of inflaion under discreion. Therefore, he inflaion bias problem under Calvo pricing is more severe han ha under Roemberg pricing for he benchmark parameers. We now urn o discuss his resul, as well as underaking a sensiiviy analysis. To explore his difference furher, we change he value of he monopolisic compeiion disorion defined by ɛ/(ɛ 1) by varying ɛ and assessing is effec on he equilibrium inflaion bias. We inerchangeable describe his measure of he monopolisic compeiion disorion as he flexible-price markup since i measures he markup ha would be observed under flexible prices. This approach is based on he fac ha he size of he inflaion bias depends on he degree of monopolisic disorion, which makes seady sae (even flexible-price) oupu inefficien and hence higher inflaion is aracive. Figures and 3 presen how he size of inflaion bias changes as he markup is varied for he Calvo and Roemberg pricing, respecively. The benchmark ɛ = 11 yields a felxible-price markup of 1.1. When ɛ decreases, he corresponding monopolisic compeiion disorion and inflaion bias increases. To illusrae he impac of he monopoly disorion on he non-lineariy, he inflaion bias for boh cases under he linear-quadraic approximaion (LQ) are also presened. The radiional linear-quadraic mehod becomes increasingly inaccurae for larger disorions. 7 Tha is, φ = (ɛ 1)θ (1 θ)(1 βθ). 14

15 Finally, we do some comparaive saics wih he model under boh pricing approaches, in order o explore how oher parameers affec he severiy of he inflaion bias problem and he sensiiviy of he resuls obained from he linear-quadraic approach. Table and Table 3 summarize he robusness oucomes for he Calvo and Roemberg pricing, respecively. In general, he inflaion bias problem is much worse under Calvo pricing. 4.3 Discussion The Average Markup and Inflaion Bias We find ha he inflaionary bias problem is significanly greaer under Calvo, especially as he monopolisic compeiion disorion is increased. A he same ime consumpion falls by more, and hours worked by less under Calvo as we increase his disorion, and he average markup rises above he flexible price markup under Calvo, while decreases under Roemberg as a resul of he non-linear effecs of he inflaion bias. See Figures and 3. In undersanding he resuls i is helpful o consider he effecs of inflaion on he wo models. Ascari and Rossi (01) discuss how inflaion affecs boh models hrough a wedge effec as well as an average markup effec. We shall consider he wedge effec firs, before urning o he average markup effecs, which will urn ou o be key. Under boh forms of nominal ineria he wedge implies ha he represenaive household s aggregae consumpion will be lower for a given level of labour inpu as inflaion rises. Under Calvo his is because he dispersion of prices means ha hey need o consume relaively more of he cheaper goods o compensae for he expensive goods given diminishing marginal uiliy in he consumpion of each good. As Damjanovic and Nolan (010a) noe his is akin o a negaive produciviy shock, where we can combine he resource and aggregae producion funcion o yield, A C = (1 + ψ c ) N where he inefficien wedge under Calvo, ψ c = 1, capures he exen o which price dispersion has been raised above one. Under Roemberg he micro-foundaions of he wedge is differen - adjusing prices uses up consumpion goods direcly. However, we can similarly combine he aggregae producion funcion and resource consrain o obain a similar expression under Roemberg, C = A (1 ψ R )N where he Roemberg wedge, ψ R = φ (Π 1) reflecs he coss per uni of oupu of changing prices. Therefore in boh cases he labour coss of aaining a paricular level of aggregae consumpion are higher, ceeris paribus, as inflaion rises. In order o assess how his affecs he inflaion bias problem facing he policy maker i is helpful o imagine how a social planner would respond o he exisence of such wedges were he o imagine hem o be exogenously given in he manner of a echnology shock. Given he form of household uiliy, he social planner would choose an opimal level of labour inpu of ( ) 1 σ N σ+ϕ A = (1 + ψ c ) under Calvo, and N σ+ϕ = ( A (1 ψ R ) ) 1 σ 15

16 under Roemberg. Therefore, for our benchmark calibraion of σ = 1 he social planner would no seek o adjus he labour inpu ino he producion process as a resul of increases in eiher of he wedges, bu would simply allow consumpion o fall. In oher words, for our benchmark calibraion he efficiencies implied by hese wedges do no give he policy maker a furher desire o generae a surprise inflaion, ceeris paribus. While if σ > 1 he social planner would seek o reduce he labour inpu as eiher of hese inefficiency wedges increased. Tha is, in his case he wedges would reduce he desire o encourage firms o employ more workers ceeris paribus. We can see his from Tables and 3 where raising he inverse of he ineremporal elasiciy of subsiuion, σ, reduces he inflaion bias under boh pricing models. Therefore he differen inefficiency wedges under Calvo and Roemberg are no responsible for he observed inflaion biases. Insead he differences in inflaion bias across he wo models are generaed by heir average mark-up behavior, which is fundamenally differen. Consider he seady-sae of he average markup (equal o he inverse of real marginal cos) under Roemberg which is obained by rearranging he deerminisic seady sae of he new Keynesian Phillips curve (NKPC) under Roemberg as, ɛ 1 mc 1 = + ɛ ] 1 (1 β) φ(π 1)Π ɛ The second erm in square brackes exiss as a combinaion of seady-sae inflaion and discouning on he par of firms (on behalf of heir owners, he represenaive household). Essenially as he firm discouns fuure profis hey also discoun fuure price adjusmen coss. As a resul in he face of ongoing inflaion, hey will op o parially delay he required price adjusmen such ha he average mark-up is decreasing in inflaion. The effec of inflaion on he average mark-up under Calvo is, mc 1 = ɛ ɛ 1 ( ) ( 1 θβπ ɛ 1 1 θπ ɛ 1 1 θβπ ɛ 1 θ ) 1 ɛ 1 In his case he effecs of inflaion on he average markup are ambiguous. However, following King and Wolman (1996) his can be decomposed ino wo elemens - he marginal markup, P MC = ɛ ( ) 1 θβπ ɛ 1 ɛ 1 1 θβπ ɛ and he price adjusmen gap, P P = ( 1 θπ ɛ 1 1 θ ) 1 ɛ 1. Here we can see ha higher inflaion raises he marginal markup. Firms facing he possibiliy of being suck wih he curren price for a prolonged period will end o raise heir rese price when ha price is likely o be eroded by inflaion hroughou he life of ha conrac. The effec of inflaion on he price adjusmen gap will end o reduce his elemen of he average markup. However, excep a very low raes of inflaion, he effecs of inflaion on he average markup hrough he marginal mark-up effec are posiive. Therefore we would expec o see average markups rise wih inflaion under Calvo, bu fall under Roemberg. This, in urn, implies ha he inflaionary bias problem is worsened under Calvo as he rising markups increase he policy makers incenives o 16

17 inroduce a surprise inflaion ceeris paribus, a he same ime as i is miigaed under Roemberg. As a resul he inflaion bias problem is significanly higher under Calvo where consumpion falls by more and hours by less han i does under Roemberg Sensiiviy Analysis Tables and 3 consider he robusness of our resuls across various parameers for Calvo and Roemberg pricing, respecively. The firs hree rows of each Table increase he degree of nominal ineria (where he Roemberg price adjusmen parameer is adjused in line wih he changes in he Calvo parameer such ha he linearized NKPC is equivalen across boh forms of nominal ineria). As we increase he degree of nominal ineria, we find ha he inflaion bias rises under Calvo, bu falls under Roemberg. This is for he reasons discussed above. Under Calvo greaer price sickiness means ha firms are likely o be suck wih heir curren price for longer, meaning ha hey aggressively raise prices when given he opporuniy o do so. This will end o raise average markups and worsen he inflaionary bias problem. In conras under Roemberg, higher price adjusmen coss resul in firms wishing o delay price adjusmen which reduces average markups and reduces he inflaion bias problem. The nex piece of sensiiviy analysis looks a various parameerizaion of he inverse of he ineremporal elasiciy of subsiuion, σ. As noed above, a he benchmark value of σ = 1, he social planner would no wish o expand employmen as eiher of he efficiency wedges due o he wo forms of nominal ineria increase. While if σ < (>) 1 hen hey would wish o increase (decrease) he labour inpu as eiher efficien wedge increased. Therefore we see he inflaionary bias falling as σ increases across boh forms of nominal ineria. Finally, we consider an increase in he inverse of he Frisch elasiciy of labour supply, ϕ, which serves o reduce he inflaionary bias problem across boh ypes of price sickiness. As labour supply becomes less elasic here is less desire o use cosly inflaion surprises o achieve only marginal increases in he level of oupu and he inflaion bias falls Relevance of Resuls In order o assess he implicaions of our calculaed levels of inflaion bias under he Roemberg and Calvo forms of nominal ineria, we conras our inflaion raes wih boh he empirical esimaes of rend inflaion and he criical values of rend inflaion a which he sandard model develops non-sandard properies. Empirical Esimaes of Trend Inflaion Cogley and Sargen (00) s esimaes of rend inflaion in a Bayesian VAR wih ime varying coefficien suggess ha a large par of he movemens in inflaion in he pos-war period (is rise in he 1970s o is fall in he 1980s) was due o he evoluion of rend inflaion raher han flucuaions around ha rend. Similarly, Cogley and Sbordone (008) find ha here is no ineria in priceseing behavior due o indexaion-ype behavior, bu ha he ineria in he daa can be described by he evoluion of rend-inflaion in a generalized NKPC. While Ireland (007) finds ha changes in he Fed s inflaion arge can help explain inflaion dynamics, where ha arge rose from 1.5% in 1959 o over 8% in he lae 1970s, before falling back o.5% in 004. Therefore, o he exen ha observed inflaion reflecs movemens in an underlying rend i would sugges ha he empirical measures of rend inflaion could easily be consisen wih our measures of he inflaionary bias wihou having o resor 17

18 o implausibly high monopolisic compeiion disorions. Moreover, when we augmen he model wih an esimaed process for mark-up shocks he magniude of he resulan inflaion volailiy can easily accoun for he observed volailiy of inflaion around is ime-varying rend. Trend Inflaion and Deerminacy In order o furher assess he implicaions of our calculaed levels of inflaion bias under he Roemberg and Calvo forms of nominal ineria, we conras our inflaion raes wih he key values of rend inflaion a which he sandard model develops issues wih he deerminacy of sandard ineres rae rules. We could, of course, have looked a oher feaures highlighed in he rend inflaion lieraure such as he learnabiliy of he model as rend inflaion rises or is impulse responses o moneary policy shocks and so on, bu since he bifurcaion in deerminacy condiions reflecs a common underlying non-lineariy which drives all he phenomena in he rend inflaion lieraure we choose o focus on his as a sraigh-forward way of assessing wheher or no our calculaions sugges he concerns raised by he rend inflaion lieraure are significan or no. Accordingly, we follow Ascari and Rossi (01) and linearize our wo economies around a deerminisic seady-sae wih an arbirary rae of seady-sae inflaion (deails of he linearized models are provided in appendix C.4). We hen assume a sandard parameerizaion of a Taylor rule for moneary policy, R = 1.5π + 0.5y, and for a range of values of he monopolisic compeiion disorion/flexible price mark-up, ɛ/(ɛ 1), we compue he seady-sae rae of inflaion a which he sandard Taylor rule flips from being deerminae o being indeerminae. We hen plo his deerminacy fronier in disorion-inflaion space along side our inflaion bias esimaes, see Figure 4. We find ha a low levels of he monopolisic compeiion disorion he inflaionary bias number lies below he deerminacy fronier - in oher words he sandard Taylor rule would remain deerminae a he raes of inflaion implied by our inflaionary bias calculaions. However, as he markup is increased he inflaion bias esimaes cross he deerminacy fronier implying ha a he raes of inflaion implied by he inflaion bias esimaes a sandard Taylor rule would be indeerminae in a log-linearized represenaion of he model. This is paricularly rue in he case of Calvo where a flexible price markup of jus over 11%, which is well wihin he range of sandard parameerizaion in he lieraure. In conras, under Roemberg he flexible-price markup needs o be double ha push us beyond he deerminacy fronier. 5 The Effecs of Cos-push Shocks In he analysis above we have focussed on he sochasic seady sae of he non-linear policy problem o reveal he exen of he inflaion bias. However, he response o shocks can also be markedly differen across he wo forms of nominal ineria. In order o explore he effec of cos-push shocks 8 on policy rade-offs under discreion in our fully nonlinear model, we, adop he esimaed shock process from Chen e al. (014) which is modelled as a revenue ax rae flucuaing around a seady sae value of zero, ln (1 τ p ) = (1 ρ τp ) ln (1 τ p ) + ρ τp ln (1 τ p 1 ) e τ 8 The echnology shocks already presen in our model do no creae meaningful policy rade-offs under our benchmark calibraion largely resuling in offseing ineres rae movemens regardless of he form of nominal ineria. 18

19 where e τ N(0, ) and ρ τp = In a log-linearized model his is equivalen o allowing for flucuaions in a desired mark-up hrough variaions in ɛ. However, in our non-linear model allowing ɛ o be ime varying has a direc impac on he measure of price dispersion in a way which would no normally be considered o be an inheren par of a cos push shock. Therefore we focus on variaions in a revenue ax a means of generaing an auocorrelaed cos push which is consisen wih he daa. The complee model wih he ime-varying revenue ax rae is presened in appendix C.5. We presen wo ses of resuls. In he firs we consider he impac of an inflaionary cos push shock wih our benchmark parameerizaion, bu wih θ = 0.65, and φ = These respecive measure of price sickiness imply an idenical seady-sae rae of inflaion of 1.95%. Figure 5 reveals ha even a his relaively modes degree of inflaion bias, here are non rivial differences in he impulse responses o an idenical cos push shock. These are driven by he same economic mechanisms observed in he seady sae analysis above as average markups rise under Calvo exacerbaing he effecs of he cos push shock. I should be noed ha he convenional way of parameerizing he Roemberg price adjusmen cos parameer such ha he slopes of he linearized Phillips curves are idenical would have implied a far lower value of φ = In fac given he significan differences in he inflaion bias across he wo forms of price-sickiness i is generally no possible o calibrae he Roemberg parameer by seeking o mimic he seady-sae rae of inflaion observed under Calvo, ceeris paribus. Therefore, in a second exercise we ensure a common seady-sae rae of inflaion of.54% by adoping he following se of parameers, ɛ = 11, θ = 0.8 under Calvo, and ɛ = , φ = under Roemberg. This calibraion ensures ha boh forms of nominal ineria generae idenical seady-sae raes of inflaion and levels of oupu. Despie sharing a seady sae in hese dimensions, he response o he idenical shock is markedly differen across Calvo and Roemberg. In Figure 6 we can see ha inflaion is 0.% higher on impac from an idenical cos push shock under Calvo, while oher variables, paricularly oupu and consumpion, exhibi a hump-shaped response o he shock due o he gradual evoluion of price dispersion, which is no a feaure of he response o he shock under Roemberg pricing. 19

20 6 Conclusion In his paper we have conrased he properies of he Calvo and Roemberg forms of nominal ineria which are commonly used in New Keynesian analyses of macroeconomic policy. They are ofen reaed as being inerchangeable, largely because hey generae equivalen NKPC and policy implicaions when linearized around an efficien zero-inflaion seady sae. However, our non-linear soluion of he discreionary policy problem reveals some sriking differences across he wo models of price sickiness. Firsly, he inflaion bias problem is far greaer under Calvo pricing han Roemberg pricing, despie he fac ha he coss of inflaion are significanly higher under he former. The reason for his is ha inflaion raises he average markup under Calvo pricing as firms seek o raise heir prices more aggressively whenever hey can o avoid he erosion of heir relaive price due o inflaion. This increase in average markups worsens he inflaionary bias problem. In conras, under Roemberg pricing firms can adjus prices in every period, and will moderae heir average markups as inflaion rises as hey aemp o delay some of he coss of price adjusmen due o he discouning inheren in heir objecive funcion. Secondly, for empirically reasonable levels of monopoly power he inflaion bias ha emerges from boh models implies ha he raes of inflaion idenified as being rend inflaion in empirical sudies are reasonable. Moreover, he raes of inflaion implied by he model are sufficien for he non-lineariies inheren in he model o place he economy in he zone where he effecs of rend inflaion are found, in sudies which approximae he economy around a non-zero rae of seady-sae inflaion, o have profound implicaions for he deerminacy properies of rules, he learnabiliy of he raional expecaions equilibirum and he ransmission of moneary policy. Tha is, he degree of inflaion bias generaed by he model implies ha he non-lineariies inheren in he model and he choice of form of nominal ineria maer. Thirdly, we exended he model o consider he sabilizaion of he economy in he face of mark-up shocks. Here we find ha he non-lineariies ha generae radically differen degrees of inflaion bias in he seady sae also imply profound differences in he moneary policy response o he same shock boh across models, wih he inflaion response o a cos-push shock being significanly greaer under Calvo, while possibly also being associaed wih a hump-shaped oupu/consumpion response as a resul of he evoluion of price dispersion which is absen from he Roemberg model and ypically ignored in he linearized New Keynesian model. 0