Concepto y definición de la historia de la filosofía

Let the central bank have a long-run socially optimal inflation target n and two

short-run goals of stabilizing output and inflation around the long-run inflation target.43 The central banks’ period loss function can be written as,

L(jz„y^) = ~ [( n, - n f + X y f - } (3.27)

A >0 represents relevant weight on output stabilization (for A =0 it becomes

single goal objective). With 0 < S < 1 as discount factor the intertemporal objective

function become

min Et ^ S T~'L(7rT, y f ) (3.28)

Subject to:

Supply function: zrf+1 = n t +cc y f + £t+l (3.29)

Demand function: y A+1 = ß yy* - ß r{it- n t ) + rjl+x (3.30)

where y f is output gap (log actual relative to potential output), n t = p t - is

inflation rate, it is monetary policy instrument (say short-term interest rate), p t is

log of price level, 77,, and et are white noise shocks in year t and are not known in

year t — 1. Coefficients a vand ß r are positive, while ß y is non-negative but less

than one. The interest rate affects output with one-year lag, and hence inflation with two-year lags, thus two year is the control lag in the model. Transmission channels include an aggregate demand channel and expectation channel. With the aggregate demand channel, monetary policy affects aggregate demand, with lag, via its effect on the short-run real interest rate and credit availability. Aggregate demand then affects inflation, with another lag, via an aggregate supply equation probably like a

43 In a more general set up a central bank can include several other variables such as exchange rate and interest rate in its loss function. However, such considerations will reduce the weight on inflation stabilization. For expositional simplicity, only two-variable loss function is considered here.

Phillips curve. The expectation channel allows monetary policy to affect inflation expectations, which, in turn, affect inflation, with a lag, via wage and price setting behavior.

Taking expectations of equation 3.30, the central bank’s reaction function can be written as follows.

1 a ß y a

h = * . - j y ^ + j y ,

(3.31)

Now, equation 3.31 will represent the optimal reaction function of the central bank if the forecasted value of output gap, y A,|f is chosen optimally. In order to find optimal y A+1|f the optimization problem is carefully formulated as follows. Equations (3.29) and (3.30) can be expressed as equations (3.32), (3.33) and (3.34), where (3.34) is obtained by advancing equation (3.29) one period ahead and using the relationship obtained in equations (3.32) and (3.33).

^ t+1 ^ f + i | / " ^ / + i V,A+1 =

y?+i\t

+ ^ + 1 n t+2\t+1 = 7 T + 1 JrCCy y f + \ - n ,+\\t + a y y t l \ t + ( £ r + l JrCCy V t+\ ) (3.32) (3.33) (3.34)

With the above transformations, the problem of dynamic program can be stated in terms of the value function as follows with one-period-ahead expected output gap

yt+l|, as the state variables.

V(*,t i|,) = + A y,t,|,2]+<5E,V(^„2H )J (3.35)

Subject to: nM +(£■„, + a,t7,+l) (3.36)

Let the conjectured solution of the above problem be as given in equation (3.37)

'/ (*,+1|,) = * „ + L ( t f ,+l| , - t f ' ) 2 (3.37)

Using the corresponding Bellman equation for the above problem and the conjectured solution of (3.37) the first order condition can be obtained as follows.

^t+2 \t

*

- n

A

ö a yk

y%

1

|,

(3.38)

Equation (3.38) is at the heart of the concept of the inflation targeting problem (Svensson 1999). It says that select the instrument such that deviation between the two-year conditional inflation forecast is --- times the negative of the one-year

Sctyk

output-gap forecast. The above relationship is derived involving both inflation and output forecast. In case only inflation forecast is involved in the targeting rule, A =0, the strict inflation targeting rule is obtained as:

x t+2\t = n (3-39)

Equation (3.39) says ‘set the instrument such that the conditional inflation forecast at the two-year horizon (or, more generally, at the shortest horizon at which inflation can be affected) equals the inflation target’ (Svensson 1999:629). Advancing (3.29) by one period and taking expectation the expected output gap in period t, the one period ahead conditional forecast of output gap can be written as

y$r l|r r+2|r (3.40)

Using equation (3.40), the first order condition (3.38) can also be written as follows:

Where c(/1) is defined as c =---— , which fulfills 0 < c < 1 and

A + S a 2yk

dc dc

— > 0, --- < 0 ,44 Now, the targeting rule of the form of equation (3.41) says ‘set

dA d a y

the instrument such that the two-year conditional inflation forecast’s deviation from the inflation target is the fraction c(A) of the one-year conditional inflation forecast’s deviation’.

Using the first order condition (3.38) and relationships presented in equations (3.40), (3.41), and (3.29), the optimal reaction function (3.31) can now be written as follows: S a yk _{l -} *

V

ß y ß r * \ t+2\t ~ n r ~ ß S a k / = 7Tr S a yk I' ) + y ' ß r 8a] k

y

+ A

A -I- S a yk y, (3.42)

K, +b,(n,

-f here bm S a yk ^ l - c ( A ) ß \ A + S a 2yk) ß ra y >0 and S a ) k A + Sa]k + ß\ 1 - c ( A ) + ß y ßr

>0. Thus, the optimal reaction function (3.42) says that the instrument

interest rate is increasing in excess of the two-year inflation forecast over the inflation target, or in excess of current inflation over the inflation target, in addition to increasing in output. Weight on output stabilization motivated a gradual adjustment of the inflation forecast towards the long-run inflation target. Less weight on output stabilization means faster adjustment towards the long-run inflation target. In the case of pure inflation targeting the reaction function reduces to the following, which is similar to the Taylor rule.

1 / l + /»v A

(3.43)

44 See (Svensson 1997) for proof and derivation of ‘k’.

In document Introducción a la Historia de la Filosofía (página 178-200)