MA Advanced Macro, 2016 (Karl Whelan) 1

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1 MA Advanced Macro, 2016 (Karl Whelan) 1 The Calvo Model of Price Rigidiy The form of price rigidiy faced by he Calvo firm is as follows. Each period, only a random fracion (1 ) of firms are able o rese heir price; all oher firms keep heir prices unchanged. When firms do ge o rese heir price, hey mus ake ino accoun ha he price may be fixed for many periods. Firms operae in an imperfecly compeiive marke, so ha if here were no fricions hey would se prices as a fixed markup over marginal cos. In mos modern applicaions, he marke srucure of he economy follows ha se ou in a famous 1976 paper by Avinash Dixi and Joseph Sigliz. The basic model has no capial, so all goods are consumpion goods, and consumers seek o maximize a uiliy funcion over a coninuum of differeniaed goods given by ( 1 Y = 0 Y (i) θ 1 θ ) θ θ 1 di This leads o demand funcions for he differeniaed goods of he form where P is he aggregae price index defined by (1) ( ) P (i) θ Y (i) = Y (2) P ( 1 P = 0 ) 1 P (i) 1 θ 1 θ di The Calvo assumpion abou price sickiness implies ha his price level can be re-wrien as which can be re-wrien as P = P 1 θ (1 ) X 1 θ = (1 ) X 1 θ ] + P 1 1 θ 1 1 θ (3) (4) + P 1 θ 1 (5) We now know ha o apply he soluion echniques for raional expecaions models, hese ype of equaions need o be log-linearized. This is done as follows. If we log-linearize his equaion around a zero-inflaion seady-sae such ha hen we have X = P = P 1 = P (6) (P ) 1 θ (1 + (1 θ) p ) = (1 ) (P ) 1 θ (1 + (1 θ) x ) + (P ) 1 θ (1 + (1 θ) p 1 ) (7)

2 MA Advanced Macro, 2016 (Karl Whelan) 2 which simplifies o p = (1 ) x + p 1 (8) Opimal Pricing in he Calvo Model How do firms ha ge o change heir price a ime se his price? They model assumes ha hey selec his opimal rese price o maximize he expeced presen discouned values of real profis over he course of his price conrac, where β is he discoun rae. This presen value is given by ( E (β) k Y +k P+k θ 1 X1 θ P 1 +k C ( )) ] Y +k P+kX θ θ (9) where C (.) is he nominal cos funcion. Noe ha appears in he discoun rae because k is he probabiliy ha his paricular price conrac is sill in exisence in k periods ime. Differeniaing his equaion wih respec o X we ge he following firs-order condiion: ( E (β) k (1 θ) Y +k P+k θ 1 X θ ) ] + θmc +k Y +k P+k θ 1 X θ 1 = 0 (10) This can be solved o give he following soluion for he opimal price ( ) X = θ E (β) k Y +k P+k θ 1 MC +k ( θ 1 ). (11) E (β) k Y +k P+k θ 1 This equaion looks very complicaed bu is quie inuiive if i is inspeced closely. Wihou pricing fricions, he opimal rule for an imperfecly compeiive firm wih elasiciy of θ demand θ is o se prices as a markup over marginal cos, such ha price equals θ 1 imes marginal cos. In his case, because he price is likely o be fixed for some period of ime, his equaion saes ha he opimal price is a markup over a weighed average of fuure marginal coss. The weigh for each fuure marginal cos has wo elemens o i: The erm (β) k which pus less weigh on fuure marginal coss because of discouning and because he price being se now has lower probabiliies of sill being around in k periods ime as k ges bigger. The erm Y +k P+k θ 1 which represens aggregae facors affecing firm demand in he fuure. As Y +k goes up he firm will sell more; as he aggregae price level P +k goes up, he firm s relaive price goes down and is demand increases. This facor will

3 MA Advanced Macro, 2016 (Karl Whelan) 3 probably somewha offse he discouning erm if he firm is going o sell a lo more in k period s ime han now, hen i may pu a bi more weigh on marginal cos in ha period. Log-Linearizing he Opimal Pricing Rule To ge he opimal pricing equaion ino a linear form ha we can use in our compuer programs, we log-linearize he firs-order condiion (10) around a zero inflaion seady-sae wih consan oupu, he firs erm inside he big curly bracke becomes (1 θ) Y +k P θ 1 +k X θ (1 θ) Y (P ) θ 1 (X ) θ (1 + y +k + (θ 1) p +k θx ) (12) while he second erm becomes θmc +k Y +k P θ 1 +k X θ 1 θmc Y (P ) θ 1 (X ) θ 1 (1 + mc +k + y +k + (θ 1) p +k (1 + θ) x ) One can hen use he fac ha in seady-sae o hen simplify his o which simplifies furher o X = (13) ( ) θ MC (14) θ 1 ] E (β) k (x mc +k ) = 0 (15) x = (1 β) (β) k E mc +k (16) The erms in aggregae oupu and he price level drop ou and he equaion says ha he log-price is a weighed average of expeced fuure logs of marginal cos. The New-Keynesian Phillips Curve So, he dynamics of pricing in he Calvo model can be summarized by he wo equaions p = (1 ) x + p 1 (17) x = (1 β) (β) k E mc +k (18)

4 MA Advanced Macro, 2016 (Karl Whelan) 4 The firs equaion can be wrien in he form required by soluion algorihms such as Binder-Pesaran. The second equaion, which involves an infinie sum, can no. However, i describes he sandard soluion o a firs-order sochasic difference equaion, and hus i can be reverse engineered o be wrien in ha forma as x = (1 β) mc + (β) E x +1 (19) Equaions (17) and (19) can be pu on he compuer and combined wih a model of marginal cos, his can be solved o describe prices and inflaion in his model. I urns, however, ha here is also a nea analyical soluion ha sheds useful insigh on how inflaion behaves in his model. This is derived as follows. Firs, noe ha equaion (17) can be re-arranged o express he rese price as a funcion of he curren and pas aggregae price levels x = 1 1 (p p 1 ) (20) Subsiuing his ino equaion (19) we ge 1 1 (p p 1 ) = (1 β) mc + β 1 (E p +1 p ) (21) Muliplying across by 1 and collecing erms, his becomes ) p (1 + 2 β p 1 = βe p +1 + (1 ) (1 β) mc (22) We can divide boh sides by and hen subrac βp from boh sides o ge ( 1 β + 2 ) β (1 ) (1 β) p p 1 = βe π +1 + mc (23) where π = p p 1 (24) is he inflaion rae. Now we can use he fac ha To obain 1 β + 2 β π = βe π +1 + Or, defining real marginal cos as = 1 + (1 ) (1 β) (1 ) (1 β) (25) (mc p ) (26) mc r = mc p (27)

5 MA Advanced Macro, 2016 (Karl Whelan) 5 he model implies π = βe π +1 + (1 ) (1 β) mc r (28) In oher words, inflaion is a funcion of expeced inflaion and real marginal cos. This equaion is known as he New-Keynesian Phillips Curve (NKPC).