When both wages and prices are sticky

Transcription

1 Whe boh ages ad rices are sicy Previously, i he basic models, oly roduc rices ere alloed o be sicy. I racice, i is ossible ha oher rices are sicy as ell. I addiio, some rices migh be more or less sicy ha ohers ad his could be imora for he dyamics i he ecoomy as a hole. For isace, roducer rices, cosumer rices, roduc rices, rices o services, ages, rices o radeables, asse rices, ec. may differ subsaially ih resec o siciess. I his secio e herefore exed he basic model o iclude sicy ages. I is ell-o ha ages, aricularly for labor ih o-sadard jobs, are sicy ad ha labor coracs may have a log duraio. I Sede, for isace, i is resely commo ih 3-year coracs. The model reseed here as origially develoed by (Erceg, C. J., D. W. Hederso ad A. T. Levi 2000) ad a someha simlified versio of heir model is described i (Gali, J. 2008), ho is folloed here. The sicy ages follo he same model from (Calvo, G. A. 983) ha as alied for roducer rices i he basic model. This meas ha here is mooolisic comeiio i he labor mare ad ha he households suly differeiaed labor o he firms, aalogous o he firms sulyig differeiaed roducs o he cosumers. Each eriod a cosa fracio of he households are alloed o chage heir omial age. Cosequely, he aggregae omial age resods sluggishly o shocs, imlyig iefficie marus o ages. Also, aalogously ih he roduc mares, relaive ages chage i resose o omial shocs ad creae a iefficie allocaio of labor. For he ceral ba, here is o he addiioal roblem of ho o bes couerac hese iefficiecies. The model ih sicy ages ad rices Firms A coiuum of firms is assumed, similar o he basic model: Y() i = AN () i () here N () i is a idex of labor iu used by firm i ad defied by ε ε ε N() i N (, ) 0 i j d j (2) here N (, i j) deoes he quaiy of he ye j labor emloyed by firm i i eriod. The arameer ε is he subsiuio elasiciy amog labor varieies. Le W ( j ) be he omial age for ye j labor i eriod. Wages are se by orers, or by uios ha rerese hem, of each ye ad hese ages are ae as give by firms. Firms miimize cos ad choose he demad of labor of each ye, give he firms oal emloyme (ouu) ad is obaied as he arial derivaive of he cos fucio ih resec o age of he jh ye of labor as ε W ( j) N(, i j) = N() i W (3)

2 here ε (, ) ε 0 W = W i j dj (4) is a aggregae age idex. Subsiuig (3) io (2) yields he age bill 0 W ( j) N (, i j) d = j WN () i as he roduc of he age idex ad he emloyme idex. The firms he solve he same roblem as i he basic model, i.e. max P = 0 {, + ( ψ + + ( + ) )} θ E Q PY Y subjec o he sequece of demad cosrais Y + P = P+ ε C + here Q, σ C P + + β C P+ is he sochasic discou facor for omial ayoffs, ψ + (.) is he cos fucio ad Y + deoes he ouu i eriod + for a firm ha las rese is rice i eriod. As i he basic model a Phillis curve ca be derived as { + } π = β π λµ (5) ˆ E ( θ )( βθ ) α here ˆ µ µ µ = ( mc mc) ad λ θ α + αε. Households As i he basic model he yical household maximizes { β ( ( ), ( )) = } E U C j N j 0 0 subjec o a sequece of budge cosrais here N ( j ) is he quaiy of labor sulied ad ε ε ε C( j) C (, ) 0 i j di (6) here i refer o he ye of good ad j refer o he ye of labor ha he household is secialized o suly.

3 I each eriod a fracio θ rese heir age, hile a fracio ee heir age fixed. Oimal age seig Cosider he household ha oimize ad chooses he age { ( βθ ) 0 (, = + + )} θ W i order o maximize E U C N (7) he execed discoued uiliy from cosumio ad leisure (disuiliy from orig) durig he eriod durig hich he age is execed o be fixed a he level W se i he curre eriod. (7) is maximized subjec o he cosrais give by he demad fucios (3) ad he flo budge cosrais ha are effecive hile W is. The oimaliy codiio ca be rie = 0 W ( βθ) E N Uc( C, N ) +ΜU( C, N ) P + here ε Μ = ε is he age maru ad be rerie as W 0 E N U c C N MRS = Μ + P+ (8) ( βθ ) (, ) here MRS + (, + + ) (, + + ) U C N is he margial rae of subsiuio beee cosumio ad U C N c hours ored i eriod + for he household ha reses is age i eriod. I he secial case of W W full age flexibiliy, θ = 0, e have = = Μ MRS. Tha meas ha he age maru is P P he edge beee he real age ad he margial rae of subsiuio i he absece of age rigidiies. Log-liearisig aroud he seady sae yields he folloig aroximae age seig rule E mrs (9) ( ) { + + } = µ + ( βθ) βθ = 0 + here µ = log Μ. The age equaio is icreasig i execed fuure rices, hich reflecs ha households are cocered abou heir urchasig oer. The age is also icreasig i he margial rae of subsiuio of labor (disuiliy of or) i erms of goods over he life of he se age, because households adjus heir execed real age, give execed fuure rices. I ca also be see as deedig o he value of he margial disuiliy of orig hours i erms of goods. As i he basic model he secificaio (, ) U CN σ + ϕ C N = σ + ϕ hich alied o (9) gives

4 ( ) { + } ( ) ( ) = βθ E + βθ + ε ϕ µ (0) ˆ here ˆ µ µ µ deoes he deviaio of he average age maru from is seady sae value µ ad ( ) mrs µ =. The log-liearised aggregae age equaio ca he be derived as = θ + ( θ) () By combiig (0) ad () ad usig π = oe obais he age iflaio equaio { + } π = β π λµ (2) ˆ E ( θ)( βθ) here λ =. The age iflaio equaio is aalogous o he rice iflaio θ( + εϕ) equaio (NKPC) i (5) ad he ierreaio of i is similar. Whe he desired maru is belo he seady sae level he age iflaio icreases i order o cach u. I his exeded model he age iflaio equaio (2) relaces he codiio = mrs i he basic model. As i he basic model here is a Euler equaio for he cosumers ( ) { + } { + } c = E c i E π ρ (3) σ here as before i he basic model i = log Q is he yield o he oe eriod bod. Equilibrium The aalysis of equilibrium maes use of ouu gas y = y y here he aural rae of ouu is he ouu ha maerializes i he absece of boh sicy rices ad sicy ages. The age ga is defied as ϖ = ϖ ϖ here ϖ = is he real age rae ad ϖ is he real age i absece of sicy rices ad ages, give by ϖ = log( α) + ψ a µ a here ψ a αψ α ya ad ψ ya + ϕ, he laer already derived i he basic model. σ( α) + ϕ+ α We ca o use µ = m ϖ o ge he rice maru ga

5 ˆ µ = ( m ϖ ) µ = ( y ) ϖ α = y ϖ α (4) Combiig he revious rice iflaio equaio ih (4) gives { + } π = βe π + κ y + λϖ (5) here κ αλ. Similarly, e ca calculae α ˆ µ = ϖ mrs µ ( y ) = ϖ σ + ϕ ϕ = ϖ σ + y α (6) for he age maru ad by combiig (2) ad (6) ge { + } π = βe π + κ y λϖ (7) ϕ here κ = λ σ +. I addiio here is a ideiy relaig he chages i he age ga o α rice ad age iflaio ad he aural real age give by ϖ ϖ + π π ϖ (8) The dyamic IS equaio is o derived as ( { π + } ) { + } = + y i E r E y (9) σ here he aural rae of ieres rae r ρ σe{ y } + is he rae i a equilibrium ih flexible rices ad ages. Fially, he model is closed by formulaig a ieres rule i = ρ+ φπ + φπ + φy + v (20) y here v is a disurbace erm. (5) (20) forms a dyamic sysem hich may or may o have a uique soluio. (Gali, J. 2008) shos ha his sysem i geeral has o uique soluio. Hoever, resricios ca be imosed o he ieres rae rule (20) ha guaraees a uique soluio. Provided φ = 0 he codiio φ + φ > guaraees a uique soluio y = π = π = 0 for all. Diagram y shos he deermiacy regios i ha case.

6 Diagram. Regios of deermiacy ad ideermiacy he φ y = 0. Source: (Gali, J. 2008). (Gali, J. 2008) also sudies he effecs of moeary olicy shocs, i.e. shocs o he ieres rae rule i (20). Gali assumes φ =.5 bu φ = φy = 0 so ha oly he effecs of he sicy ages are sudied. He also assumes θ = 2/3 ad θ = 3/4. I diagram 2 he effecs of he shoc, hich is a 0.25 erceage ois icrease i he exogeous comoe of he ieres rae rule, v, hich i he absece of edogeous comoes i he rule ould corresod o a oe erceage oi icrease i he aualized omial ieres rae. The solid lie shos he resose he boh age ad rices are sicy ad he dashed lies he rices or ages are sicy, resecively. Obviously he, age iflaio shos a large imac he ages are flexible ad rice iflaio he rices are flexible. Hoever, he mos realisic resose seems o be realized he boh rices ad ages are flexible.

7 Diagram 2. Sicy rices ad/or ages ad he effecs of moeary olicy shocs. Source: (Gali, J. 2008). Moeary olicy ih sicy rices ad ages A olicy ha sees o sabilize rices oly is suboimal he ages are sicy Some secial cases Simulaios ih simle rules

8 ) Refereces CALVO, G. A. (983): "Saggered Prices i a Uiliy-Maximizig Frameor," Joural of Moeary Ecoomics, 2, ERCEG, C. J., D. W. HENDERSON, ad A. T. LEVIN (2000): "Oimal Moeary Policy ih Saggered Wage ad Price Coracs," Joural of Moeary Ecoomics, 46, GALI, J. (2008): Moeary Policy, Iflaio, ad he Busiess Cycle. Priceo: Priceo Uiversiy Press.