Deriving (9.21) in Walsh (1998)

Transcription

1 Deriving (9.21) in Wlsh (1998) Henrik Jensen April8,2003 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 : nd y t y t + v t m t + π t + p t1 + π + u t,! y t 1+ " v t + m t π t p t1 + π + u t, y t + v t + m t π t p t1 + π + u t, 1

2 y t + (m t π + (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, from whih we get the solution for the infltion rte for given money supply: + (m t π # π t (π + u t ) (π t π )+z t, + (m + (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 ( + )+ nd ( + )+ (m ( + )+ u t ( + ) ( + )+ z t, ( + )+ ( + ) π ( + )+ ( + )+ (m t p t1 v t ) + ( + )+ u ( + ) t ( + )+ z t, π (1 ) ( + )+ ( + )+ (m ( + )+ u ( + ) t ( + )+ z t, 2

3 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 'm 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 'i t, follows from the LM urve s m t i ' t π t i ' t + p t1 + y t i ' t 'i t + v t. (*) Note tht we hve tht 'i t π + 1 (ρ uu t1 ρ z z t1 ) (9.17) nd π t i ' t π + ϕ t e t (9.18) We n then find y t i ' t by inserting π t i ' t into the Lus supply shedule: # y t i ' t π + ϕ t e t π + z t ϕ t e t + z t ϕ t + ρ z z t1 Then insert the found expressions for π t i ' t, y t i ' t nd 'i t into (*): m t i ' 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 'm t ρ v v t1 p t1 +(1) π ρ uu t1 + + ρ zz t1, whih pplied on (**) yields m t i ' t 'm t ρ v v t1 + v t + 1+ ϕ t 1 e t ((9.20)) 'm t + ψ t + 1+ ϕ t 1 e t 3

4 Now, when tul m t onditionl on 'i t hnges reltive to 'm t,itistime to hnge i t suh tht m t 'm t gin. Wht vlue of the interest rte will omplish tht? I.e., how do we derive eqution (9.21) on pge 399 in Wlsh (1998)? The trik is to solve the model for m t s funtion of ny vlue of the interest rte, nd then find the interest rte tht delivers m t 'm 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 'm t requires tht we use (9.19) nd find the vlue of i t tht seures this: or, (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, (1 + )+ i t π + 1+ u t + v t 1 z t ρ v v t1 +(1 ) π ρ uu t1 + + ρ zz t1, 4

5 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 # ϕ t + ψ t (1 + )+ i t # ϕ t + ψ t (1 + )+ i t i t (1 + )+ whih finlly gives π + 1+ ρ uu t1 i t % 1+ π + + # ρ z z t1 1 e t + 1 % 1+ π + + # 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, i t π + 1 (ρ uu t1 ρ z z t1 ) + (1 + ) ϕ t e t + ψ t. (1 + )+ 5

6 Using the result for 'i t, eqution (9.17), this redily redues to i t 'i t + (1 + ) ϕ t e t + ψ t (1 + )+ i T t whih is eqution (9.21) in Wlsh (1998). 6