1.3 Nominal rigidities

Transcription

1 1.3 Nominal rigidities two period economy households of consumers-producers monopolistic competition, price-setting uncertainty about productivity

2 preferences t=1 C it is the CES aggregate with σ > 1 Ã! 2X β t κ 1+η log C it N it C it = 1+ η Ã Z! σ 1 σ 1 σ 1 0 C ijt σ di,, Technology Y it = A t N it.

3 productivity shocks A t At = e a t a 1 = x + 1, a 2 = x + 2 x and t mean-zero, i.i.d., normal A signal about long-run productivity s = x + e

4 nominal balances with central bank at nominal rate R household set P it then consumers buy intertemporal BC (P 2 C i2 P i2 Y i2 )+R (P 1 C i1 P i1 Y i1 ) 0, P t is the price index P t = µz 1 P it 1 σ di 1 σ.

5 Flexible price equilibrium period 2. Optimality for price-setting, 1 P it Y it 1 Y it η (1 σ) + κσ N it =0. P t C it P it A t P it symmetric equilibrium, Y t = A t N t, this condition gives (normalization of κ). µ 1 σ 1 1+η N t = =1 κσ quantities C t = Y t = A t.

6 what about consumers decisions? consumer Euler equation " # 1 P 1 1 = RE a 1,s C 1 P 2 C 2 C t = A t log-normal 1 r + p 1 E [p 2 a 1,s]=E [a 2 a 1,s] a 1 Var[a 2 a 1,s]. 2 all changes in E [y 2 ] go to the real interest rate notice role of p 1 : neutralizes r

7 Fixedpricesinperiod1 price-setting before any shock observed " # 1 P i1 Y i1 1 η Y i1 E (1 σ) + κσ N i1 =0. P 1 C i1 P i1 A 2 P i1 rearranging this gives h i 1+η E N 1 =1 this will pin down averages but not responses to shocks

8 quantities: equilibrium in period 2 identical in period 1 now Euler equation (set p 2 =0) 1 c 1 = E [a 2 a 1,s] Var[a 2 a 1,s] r p 1. 2 suppose r fixed, p 1 fixed by assumption now sentiment shocks affect consumption

9 Figure 8: RBC and Simple Monetary Model Expectation of Technology Shock in Period 13 Not Realized Image removed due to copyright restrictions.

10 pin down p 1 h i h i 1+η E N 1 = E e (1+η)(y 1 a 1 ) =1, thanks to log-normality this equation can be solved explicitly and gives r p 1 1 Var[a 2 a 1,s]+ 1 (1 + η)(β + δ 1) 2 σ 2 x (1 + η)(β 1)2 σ (1 + η) δ2 σ 2 e =0 where E [a 2 a 1,s]= βa 1 + δs

11 simple implication anticipated changes in r are neutral if instead we follow rule, e.g. then economy response changes r = α 0 + α 1 y 1 we llgobacktomonetarypolicy

12 What about demand shocks? herepriceresponseisabsent need a bit more flexibility sticky prices imperfect information

13 1.4 Lucas-Phelps islands Lucas 1972 Overlapping generations Agents work at date t consume at date t +1 Preferences E C i,t+1 1 N 2 2 i,t

14 money

15 x t proportional subsidy from gov t agentswork,accumulatemoney,spend,die Y i,t = N i,t M i,t+1 = P i,t Y i,t (1 + x t+1 ) P j,t+1 C i,t+1 = M i,t+1 at date t +1 agent i consumes the output of agent j

16 continuum of islands, i [0, 1] unit mass of agents on each old agents receive proportional transfer x t from govt they travel to one island where they spend all their money prices P i,t determined in walrasian equilibrium young agents decide their labor supply only observe P i,t

17 old agents in island i are representative sample but different mass φ i,t nominaldemanddemandinisland i is Z 1 M i,t di = φ i,t M t φ i,t 0 φ i,t log-normal with Z 1 φi,t di = 1 0

18 Idiosyncratic demand shock log φ i,t = u i,t Monetary shocks log-normal t =log (1 + x t ) total nominal demand is D i,t = φ i,t (1 + x t ) M t 1 in logs d i,t = t + u i,t + m t 1

19 Market clearing P i,t N i,t = φ i,t (1 + x t ) M t 1

20 Information structure all agents observe {M t 1,M t 2,...} old agents observe x t,p j,t (j is the good they buy) young agents observe P i,t observing P i,t and M t 1, and knowing their own N i,t young agents can infer φ i,t (1 + x t )

21 Labor supply Agents solve max N i,t,c i,t+1 E Substitute C i,t+1 and obtain 1 C N 2 i,t+1 2 i,t P i,t,m t 1 s.t. P j,t+1 C i,t+1 = P i,t N i,t (1 + x t+1 ) FOC " # P i,t E (1 + x t+1 ) N i,t P i,t,m t 1 =0 P j,t+1

22 interpetation P i,t {z} P j,t+1 labor supply {z } exp.infl. N i,t = E[ (1 + x t+1 ) P i,t,m t 1 ]

23 Equilibrium prices guess: ³ P i,t = g φ i,t (1 + x t ) M t 1 Because φ i,t and x t are i.i.d. the distribution of φ j,t+1 (1 + x t+1 ) is given at date t. Decompose P i,t N i,t = E[ (1 + x t+1 ) P i,t,m t 1 ]= P " j,t+1 # " # P i,t M t = E i,t E i,t (1 + x t+1 ) M t P j,t+1

24 then E ³ 1+ x t+1 φ j,t+1 (1 + x t+1 ) = ξ g φ j,t+1 (1 + x t+1 ) is a constant independent of today s shocks " # P i,t 1 N i,t = ξe i,t M t 1 1+ x t

25 From equilibrium condition we obtain in logs, M t φ i,t P i,t = N i,t = ξe i,t " # P i,t M t m t p i,t + u i,t =(...) E i,t m t p i,t (constant terms in (...), depend on variances) h i We obtain 1 1 p i,t = p + (m t + u it )+ E i,t [m t ] 2 2

26 Agents observe m t + u i,t = m t 1 + t + u i,t Define E t [m t ]= Then, averaging, we have Z 1 h i 0 E m t m t 1, t + u i,t di 1 1 p t = p + m t + E t [m t ] 2 2

27 Imperfect information E t [m t ] =6 m t in particular h i E m t m t 1, t + u i,t = m t 1 + β ( t + u it ) where so β = σ 2 m σ 2 m + σ 2 u E t [m t ]=m t 1 + β t =6 m t 1 + t

28 We have 1 p t = p + m t (1 + β) t and output is y t = m t p t 1 = ȳ + (1 β) t 2 larger σ σ m 2 2 implies smaller real effects of monetary policy u Phillips curve depends on the monetary regime

29 Wrapping up with partially revealing prices first order expect. m t =6 E t [m t ] this can explain short-run non-neutrality prices adjust less than 1:1 with imp. info policy regime affects inference and thus effects of shocks