Deriving (9.21) in Walsh (2003)

Transcription

1 Deriving (9.21) in Wlsh (2003) "Monetry Eonomis: Mro Aspets" Institute of Eonomis, University of Copenhgen Henrik Jensen Mrh 24, 2004 Abstrt This note shows how to derive the nominl interest rte seuring tht the tul money supply lwys is equl to the vlue seuring the verge infltion trget. I.e., the nominl interest rte tht uses the tul money supply s n intermedite trget. The model is given by y t (π t E t1 π t )+z t y t (i t E t π t+1 )+u t m t p t m t π t p t1 y t i t + v t Wht money supply would give n infltion trget of π? The trik is to knowledge tht with the strit infltion trgeting preferenes, in expeted vluewehveinfltion on trget. I.e., E t1 π t E t π t+1 π. Hene the model is rewritten s y t (π t π )+z t y t (i t π )+u t m t p t m t π t p t1 y t i t + v t Now, the LM urve is inserted into the IS urve to eliminte i t : y t y t + v t m t + π t + p t1 + π + u t, 2004 Henrik Jensen. This doument my be reprodued for edutionl nd reserh purposes, s long s the opies ontin this notie nd re retined for personl use or distributed free. 1

2 nd y t 1+ v t + m t π t p t1 + π + u t, + y t v t + m t π t p t1 y t + (m t π + π + u t, + (π + u t ). We then find the tul infltion rte by ombining this expression with the modified Lus supply shedule: + (m t π + (π + u t )(π t π )+z t, fromwhih wegetthesolution fortheinfltion rte for given money supply: + (m t π + π t µ + + (m + (π + u t ) (π t π )+z t, + (π + u t )+π z t ( + )+ π t + + (m + (π + u t )+π z t nd therefore π t ( + )+ (m ( + )+ (π + u t )+ ( + ) π ( + ) z t ( + )+ We then solve for the vlue of m t tht seures π t π. I.e., this vlue must stisfy π ( + )+ (m ( + )+ (π + u t )+ ( + ) π ( + ) z t, ( + )+ from whih we get π + ( + ) 1 ( + )+ ( + )+ (m ( + )+ u t ( + ) ( + )+ z t, nd ( + )+ ( + ) π ( + )+ 2 ( + )+ (m t p t1 v t ) + ( + )+ u ( + ) t ( + )+ z t,

3 π (1 ) ( + )+ ( + )+ (m ( + )+ u ( + ) t ( + )+ z t, nd finlly m t p t1 + v t +(1) π u t + + z t. As shoks re unobservble, the optiml trget of m t is given by bm t p t1 + ρ v v t1 +(1) π ρ uu t1 + + ρ zz t1 (9.19) The tul money supply, for given interest rte bi t, follows from the LM urve s m t i bt π t i bt + p t1 + y t i bt bi t + v t. (*) Note tht we hve tht nd We n then find y t y t bi t π + 1 (ρ uu t1 ρ z z t1 ) (9.17) π t i bt π + ϕ t e t (9.18) by inserting π t i bt into the Lus supply shedule: µ π + ϕ t e t ϕ t e t + z t π + z t ϕ t + ρ z z t1 Then insert the found expressions for π t, y t nd bi t into (*): m t π + ϕ t e t + p t1 + ϕ t + ρ z z t1 π + 1 (ρ uu t1 ρ z z t1 ) + v t (1 ) π + p t1 + v t + 1+ From (9.19) note tht ϕ t ρ uu t1 1 e t + + ρ zz (**) t1 bm t ρ v v t1 p t1 +(1 ) π ρ uu t1 + + ρ zz t1, 3

4 whih pplied on (**) yields m t i bt bm t ρ v v t1 + v t + 1+ ϕ t 1 e t ((9.20)) bm t + ψ t + 1+ ϕ t 1 e t Now, when tul m t onditionl on bi t hnges reltive to bm t,itistime to hnge i t suh tht m t bm t gin. Wht vlue of the interest rte will omplish tht? I.e., how do we derive eqution (9.21) on pge 443 in Wlsh (2003)? Thetrikistosolvethemodelform t s funtion of ny vlue of the interest rte, nd then find the interest rte tht delivers m t bm t. This n be omplished by the entrl bnk, s it observes m t even though it doesn t observe the vrious period-t disturbnes. As the model is y t (π t π )+z t y t (i t π )+u t m t p t m t π t p t1 y t i t + v t we first ombine the AS nd IS urve to find infltion s funtion of the interest rte: (π t π )+z t (i t π )+u t nd thus π t + π i t + 1 (u t z t ) We hve output funtion of the interest rte diretly from the IS urve: y t (i t π )+u t We n the use this in the LM reltionship to find m t + π i t + 1 (u t z t )+p t1 (i t π )+u t i t + v t (1 + )+ i t π + p t u t + v t 1 z t Seuring tht m t bm t requires tht we use (9.19) nd find the vlue of i t tht seures this: (1 + )+ i t π + p t u t + v t 1 z t p t1 + ρ v v t1 +(1 ) π ρ uu t1 + + ρ zz t1, 4

5 or, (1 + )+ i t π + 1+ u t + v t 1 z t ρ v v t1 +(1 ) π ρ uu t1 + + ρ zz t1, An thus (1 + )+ i t ϕ t + ψ t 1 ρ zz t1 1 e t (1 ) π ρ uu t1 + + ρ zz t1, (1 + ) ϕ t + ψ t (1 ) π µ ϕ t + ψ t (1 + )+ i t µ ϕ t + ψ t (1 + )+ i t i t (1 + )+ π + 1+ ρ uu t1 i t π + + µ ρ z z t1 1 e t π + + µ π + ρ u u t1 ρ z z t1 1 e t ρ u u t1 (1 + )+ ρ u u t1 + ( + ) ρ z z t1 1 e t (1 + )+ π (1 + )+ + (ρ u u t1 ρ z z t1 ) + 1+ ϕ t + ψ t 1 e t, 5

6 whih finlly gives i t π + 1 (ρ uu t1 ρ z z t1 ) + (1 + ) ϕ t e t + ψ t. (1 + )+ Using the result for bi t, eqution (9.17), this redily redues to i t bi t + (1 + ) ϕ t e t + ψ t (1 + )+ i T t whih is eqution (9.21) in Wlsh (2003). 6