Different assumptions in the literature: Wages/prices set one period in advance and last for one period

Transcription

1 Øisein Røisland, Lecure 8: Moneary policy in New Keynesian models: Inroducing nominal rigidiies Differen assumpions in he lieraure: Wages/prices se one period in advance and las for one period Saggering (Taylor conracs) wages/prices se for wo periods, where one half of he conracs negoiaed each period Calvo assumpion The oppuruniy o adus he price has a given probabiliy Roenberg assumpion Quadraic adusmen coss. Firms can adus prices frequenly, bu he cos is greaer he more he price is changed. => gradual price adusmens

2 Wages se one period in advance Model: Producion funcion. Labour only variable inpu. (.) y = ( α) n + e e is whie noise produciviy shock. Labour demand: (.2) n = y ( w p) Labour supply: (.3) n = ϕ( w p ) μc Resource consrain (no invesmens): (.4) c = y 2

3 Flexible wages Equaions (.) o (.4) deermine n, y, c, and (w-p) Equilibrium wage under flexible wages: (.5) f μ w = p+ e + αϕ + μ( α) Sicky wages Assume ha nominal wages are se one period in advance. Wage seers ry o hi he equilibrium wage. w = E w = E p f (.6) since E - e =. Equaion (.2) ino eq. (.) gives (.7) y = ( w p ) + e α α α Eq. (.6) ino (.7): (.8) y = ( p E p ) + e α α α 3

4 Leingπ p p we can wrie i as a Phillips curve : (.9) Resul: π = E π + y + e α α α The governmen can affec oupu one period hrough bringing abou unanicipaed changes in he price level (e.g., hrough money supply). An unanicipaed increase in he price level reduces he real wage, and he firms demand more labour and increase producion. Weakness: No persisence in he price level. Saggering (Taylor conracs) gives persisence in he price level (bu no in inflaion) The New Keynesian Phillips curve - Monopolisic compeiion - Flexible wages - Calvo assumpion on price seing: Probabiliy of ( ω ) ha a firm is allowed o adus he price a given period (Walsh ch : Hybrid Calvo/Roemberg assumpion) 4

5 Assume ha firms face he following loss funcion: (.) 2 i * 2 β ( i, + + ) E p p where p * is he opimal price under perfec price flexibiliy. For he opimal price se oday, given nominal rigidiy, we focus on he par of he sum ha includes p, i.e., (.) ωβ ( p p ) * 2 i, + The firs-order condiion is (.2) p * i, ωβ ωβep+ = Noe ha ωβ = ωβ. Since firms are idenical, we can drop he subscrip i, and solve for he opimal price from (.2): (.3) p op * = ( ωβ ) ω β Ep+ Lead (.3) one period and ake he expecaion: 5

6 (.4) op * + = ( ωβ ) ω β Ep ++ Ep Add he erm op * + ( ) Ep + + ωβ Ep ωβ ωβ ω β o he righ-hand side of (.3). (Noe ha his erm is zero according o (.4)). Tha gives: (.5) p = ( ωβ ) p + ωβ E p + op * op Since he probabiliy ha a firm can adus he price is ( ω ), he aggregae price level reflecs ha a fracion ( ω ) of he firms have se an opimal price, and he res are suck wih he old price: op (.6) p = ( ω) p + ωp Leading (.6) one period and aking he expecaion gives: op (.7) E p + = ( ω) E p + + ω p op Solve (.7) for Ep + and inser in (.5), and inser his expression ino (.6). This gives 6

7 (.8) * 2 p = ( ω)( ωβ) p + ωβe p ω βp + ωp + Under monopolisic compeiion opimal flex-price is se as a mark-up over marginal coss: (.9) P * ε = MC ε where ε is he elasiciy of subsiuion beween various goods. In logs: p * = μ + ln( MC ) where μ =. We have ha ln( MC ) = p + mc, ε where mc denoes he log of real marginal coss. Thus, we have ha ln ε (.2) p = μ + p + mc * (.2) ino (.8) gives, afer some rearranging, ( ω)( ωβ ) p p = β ( Ep p) + ( mc + μ) ω (.2) + or, if we use ha π p p : (.22) π = βeπ+ + κmc + ν 7

8 where κ ( ω)( ωβ ) β and ν ( ω)( ωβ ) μ β If we assume ha π = in seady-sae, we can wrie (.22) as π = βeπ + κmc (.23) + where mc is he deviaion of real marginal coss from seady-sae real marginal coss. I can be shown ha in he sandard model, he oupu gap, y, is equal o mc. In addiion, i is common o add a cos-push shock, which could be inerpreed as sochasic variaions in firms marke power. We hen have he New Keynesian Phillips curve (NKP): π = βeπ + κy + u (.24) + where u is he cos-push shock. Properies of he NKP: The slope of he Phillips curve depends on he degree of price rigidiy. Slope (κ) decreases in ω. Expecaions maer for inflaion oday. Solving (.24) forward gives 8

9 (.25) π = E β y+ + u where we have assumed ha u is whie noise. There is persisence in he price level, bu no in inflaion. Appears o be in conflic wih daa. Common o include a erm wih lagged inflaion o improve he empirical fi of he model: π = ρπ + ( ρ ) β Eπ + κ y + u (.26) + Can be moived by price indexaion or rule-ofhumb behaviour. 9