Exercise, May 23, 2016: Inflation stabilization with noisy data 1

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Exercise, May 23, 2016: Inflation stabilization with noisy data 1"

Βηθεσδά Δράκος
5 χρόνια πριν
Προβολές:

1 Monetay Policy Henik Jensen Depatment of Economics Univesity of Copenhagen Execise May : Inflation stabilization with noisy data 1 Suggested answes We have the basic model x t E t x t+1 σ 1 ît E t π t+1 n t σ > 0 1 π t βe t π t+1 + x t 0 < β < 1 > 0 2 whee x t is the output gap π t is the inflation ate î t is the nominal inteest ate in absolute deviation fom the zeo-inflation steady state and t n is the natual ate of inteest in absolute deviation fom steady state. The natual ate of inteest is assumed to follow an exogenous pocess t n ρ t 1+ε n t with ε t being white noise and ρ [0 1. Inflation is measued with some i.i.d. eo t i.e. obsevable inflation o the inflation ate measued in eal time is given by π o t π t + t. 3 The cental bank follows a Taylo-type nominal inteest-ate ule based on obsevable inflation i.e.: î t φ π π o t φ π > 1. 4 i Solve fo inflation and output gap unde the ule 4 [Hint: Conjectue that the solution fo a geneic vaiable z t has the fom z t ψ z t + ψ z n t ] ANSWER: Substitute the definition of obsevable inflation 3 into the ule 4 and inset it into 1: x t σ 1 φ π π t + t E t π t+1 n t + E t x t+1 * We now conjectue the following solutions assuming that φ π > 1 secues uniqueness: Theefoe π t ψ π t + ψ π n t x t ψ x t + ψ x n t π t+1 ψ π t+1 + ψ π n t+1 x t+1 ψ x t+1 + ψ x n t+1 1 This execise builds heavily on Execise 4.1 in Jodi Galí 2008 Chapte 4 p. 90: Monetay Policy Inflation and the Business Cycle: An Intoduction to the New Keynesian Famewok Pinceton Univesity Pess. 1

2 and thus E t π t+1 ψ π E t t+1 + ψ π E t n t+1 E t x t+1 ψ x E t t+1 + ψ x E t n t+1. Using the stochastic popeties of t and n t this becomes E t π t+1 ψ π ρ n t E t x t+1 ψ x ρ n t. We now inset the conjectues and thei expected futue countepats into the model [as given by * and 2] ψ x t + ψ x t n σ [ ] 1 φ π ψπ t + ψ π t n + t ψπ ρ t n t n + ψx ρ t n ψ π t + ψ π n t βψ π ρ n t + ψ x t + ψ x n t As these equations must hold fo all ealizations of t equations to detemine the fou unknown coeffi cients: and n t we ecove the fou ψ x σ 1 φ π ψ π φ π σ 1 ψ x σ 1 φ π ψ π + σ 1 ψ π ρ + σ 1 + ψ x ρ ψ π ψ x ψ π βψ π ρ + ψ x. Fist we use ψ π ψ x ψ x σ 1 φ π ψ π φ π σ 1 to get ψ π ψ x ψ x σ 1 φ π 1 ψπ ψ π φ π 1 ψπ ψ π 1 + φ π φ π Then we use ψ π ψ x φ π > φ π ** σ 1 φ π > φ π *** ψ π βψ π ρ + ψ x ψ x σ 1 φ π ψ π + σ 1 ψ π ρ + σ 1 + ψ x ρ 2

3 ψ π 1 βρ ψ x ψ x 1 ρ σ 1 φ π ψ π + σ 1 ψ π ρ + σ 1 ψ π 1 βρ ψ x σψ x 1 ρ ψ π φ π ρ + 1 So σ ψ π 1 βρ ψ π [ σ 1 βρ 1 ρ 1 ρ ψ π φ π ρ + 1 ] + φ π ρ 1. o ψ π ψ x [ 1 σ 1 βρ 1 ρ + φ π ρ ] 1 βρ [ ] σ1 βρ 1 ρ + φ π ρ ψ π ψ x > 0 1 βρ 1 ρ + φ π ρ **** σ 1 1 βρ > 0. 1 βρ 1 ρ + φ π ρ ***** Hence the solutions fo the output gap and inflation ae x t σ 1 φ π σ 1 1 βρ 1 + φ t + π 1 βρ 1 ρ + φ π ρ n t π t σ 1 φ π 1 + φ t + π 1 βρ 1 ρ + φ π ρ n t. ii Find the solution fo the nominal inteest ate and conside the effects of φ π exteme obseved inflation stabilization on the nominal inteest ate output gap and actual inflation. ANSWER: The nominal inteest ate is [by 3 and 4] î t φ π π t + t. Inseting the solution fo π t gives [ ] î t φ π σ 1 φ π 1 + φ t + π 1 βρ 1 ρ + φ π ρ n t + t φ π 1 φ π σ + φ t + φ π π 1 βρ 1 ρ + φ π ρ n t φ π 1 + φ t + φ π π 1 βρ 1 ρ + φ π ρ n t. 3

4 In the case of φ π we theefoe find lim î t σ t + t n. We see that the nominal inteest ate moves one fo one with vaiations in the natual ate of inteest and fluctuates with the measuement eo. To see the implications fo the output gap and inflation we find the limits of the coeffi cients in the solution: lim ψ π 1 lim x 1 lim π 0 lim x 0. So the output gap and inflation become espectively lim t 1 t lim t t. Hence when φ π the measuement eo gets full powe and is tansmitted fully onto actual inflation dπ t /d t 1 consistent with dx t /d t 1.This is also consistent with the deived change in the nominal inteest ate; σ/. On the othe hand fluctuations in the natual ate of inteest ae fully stabilized in tems of thei effects on both the output gap and inflation the divine coincidence applies. This is consistent with the finding that in this limit the nominal inteest ate tacks the natual ate. iii Find the value of φ π that minimizes actual inflation vaiance in the deivation of this ignoe fo simplicity the φ π > 1 estiction. Intepet the solution. ANSWER: The solution to this poblem will depend on the elative vaiances of the measuement and natual ate shocks. High φ π is good fo natual ate shocks as it make the allocation appoach the effi cient one by having the inteest ate tacking the natual ate of inteest while a high φ π is bad in tems of measuement eos as we saw above. Inflation vaiance is given by va [π t ] ψ π 2 σ 2 + ψ π 2 σ 2 whee σ 2 and σ2 ae the unconditional vaiances of the measuement shock and natual ate shock pocesses espectively and whee it is used that these ae uncoelated. So we need to solve min φ π va [π t ] The fist-ode condition is ψ π ψ π σ 2 + ψ π ψ π σ

5 The tade off mentioned is evident as ψ π 1 + φ π 2 > 0 ψ π 2 [1 βρ 1 ρ + φ π ρ ] 2 < 0. I.e. the pat of inflation vaiance that is attibutable to measuement eos inceases with φ π while the pat attibutable to natual ate shocks ae deceasing in φ π. The optimal value of φ π must satisfy the fist-ode condition which is ewitten as ψ π 1 + φ π 2 σ2 ψ π 2 [1 βρ 1 ρ + φ π ρ ] 2 σ2 o inseting ψ π and ψ π φ π 1 + φ π 1 + φ π 2 σ2 2 1 βρ 1 ρ + φ π ρ [1 βρ 1 ρ + φ π ρ ] 2 σ2 This cannot be solved in closed fom. Note howeve that fo the special case of ρ 0 we get φ π 1 + φ π 1 + φ π 2 σ2 and theeby the optimal value of φ π as φ π σ 2 σ 2 φ π σ2. σ φ π 1 + φ π 2 σ2 This confims the intuition about how the elative impotance of eithe shock affects the value of φ π that best stabilizes inflation. Moe volatility of the natual ate calls fo a highe value of φ π and vice vesa fo moe volatility of the measuement eo. 5

Σχετικά έγγραφα

Space Physics (I) [AP-3044] Lecture 1 by Ling-Hsiao Lyu Oct Lecture 1. Dipole Magnetic Field and Equations of Magnetic Field Lines

Space Physics (I) [AP-3044] Lecture 1 by Ling-Hsiao Lyu Oct Lecture 1. Dipole Magnetic Field and Equations of Magnetic Field Lines Space Physics (I) [AP-344] Lectue by Ling-Hsiao Lyu Oct. 2 Lectue. Dipole Magnetic Field and Equations of Magnetic Field Lines.. Dipole Magnetic Field Since = we can define = A (.) whee A is called the