LEO. Stensson/ European Economic Review 46(2002)771-780 777 output gap), and setting these equal. a marginal increase in inflation two periods ahead only, d+2. />0, d+j t=0,j#2, by the aggregate-supply relation(2.6) requires a fall in the output gap one period ahead, dx+1,=-dTt+2. t/ar <0, and an equal increase in the output gap two periods ahead, dx: +2 /=-dxu+lt>0. We can then define the marginal rate of transformation of the linear combination xu+1. t F (xr+1 1, x +2,,)s (l,-lru+l, into I,+2r, MRT(,+2/,xu+1. 1), which will equal MRT(x+2,x+1)≡drx+21 From the loss function (2.1)(when the forecasts enter as arguments) follows that the marginal rate of substitution of I(+2. for xuit is given by MRS(T,+2.,xu+i,) d+2. /dx +j.lds =0=-ix1+; /(T,+2, 1-T )(in the limit when 8-1, for simplicity) From this it is easy to show that the marginal rate of substitution of I,+2, for the above linear combination xi+lt, MRS(T:+2.1, xt+1 ,), will be given by d MRS(x1+21,+1)多x+1ldg=0.dx+2=-dx (x+2t-x1+1.r) Redoing this for +r. for all t> I and setting the marginal rates of transformation equal to the marginal rates of substitution leads to the optimal specific targeting rule, *=--(x1+t-x1+t-1 where xi.t for t= l is understood to be x.i-I, the one-period-ahead forecast of the output gap in period t-1. Thus, the optimal targeting rule in this example can be expressed as"find an instrument-rate path so the inflation-gap forecast is -i/or times the change in the output-gap forecast".6 In this example, optimal"inflation-forecast targeting"can then be described as fol- lows:(1)Conditional on the judgment z E 24+,oeo, find inflation and output gap forecasts, T' E I+ee and x' E xu+r,oe, that fulfill the specific ing rule(2.8)and the aggregate-supply relation (2.6).(2)Conditional on the ment and these forecasts, find the instrument-rate forecast, i'= i+.iJeeo fulfills the aggregate-demand relation(2.7).(3)Announce these forecasts and set instrument-rate accordingly. This results in the optimal instrument-rate setting, condi- judgment, z, without having to specify As is shown in Svensson(2001b), even for this relatively simple model, the optimal reaction function is overwhelmingly complex, especially since it must specify how to respond optimally to judgment, making verifiability and commitment directly to the optimal reaction function completely unrealistic 6 As is explained in Svensson(2001b),(2.8)also applies for t=l, when x, is interpreted to be x,i-I Formulating the targeting rule this way leads to"optimality in a time-less perspective" corresponding to a ituation of commitment to optimal policy far in the past, as discussed in Woodford (1999)and Svensson d Woodford(1999)L.E.O. Svensson / European Economic Review 46 (2002) 771 – 780 777 output gap), and setting these equal. A marginal increase in in ation two periods ahead only, dt+2;t ¿ 0; dt+j;t = 0; j = 2, by the aggregate-supply relation (2.6) requires a fall in the output gap one period ahead, dxt+1;t=−dt+2;t=x ¡ 0, and an equal increase in the output gap two periods ahead, dxt+2;t = −dxt+1;t ¿ 0. We can then de@ne the marginal rate of transformation of the linear combination ˜xt+1;t ≡ (xt+1;t; xt+2;t) ≡ (1; −1)xt+1;t into t+2;t; MRT(t+2;t; x˜t+1;t), which will equal MRT(t+2;t; x˜t+1;t) ≡ dt+2;t dxt+1;t d xt+2; t=−d xt+1; t = −x: From the loss function (2.1) (when the forecasts enter as arguments) follows that the marginal rate of substitution of t+2;t for xt+j;t is given by MRS(t+2;t; xt+j;t) ≡ dt+2;t=dxt+j;t|dLt=0 = −xt+j;t=(t+2;t − ∗) (in the limit when  → 1, for simplicity). From this it is easy to show that the marginal rate of substitution of t+2;t for the above linear combination ˜xt+1;t; MRS(t+2;t; x˜t+1;t), will be given by MRS(t+2;t; x˜t+1;t) ≡ dt+2;t dxt+1;t dLt= 0; d xt+2; t=−d xt+1; t = (xt+2;t − xt+1;t) t+2;t − ∗ : Redoing this for t+;t for all  ¿ 1 and setting the marginal rates of transformation equal to the marginal rates of substitution leads to the optimal speci@c targeting rule, t+;t − ∗ = −  x (xt+;t − xt+−1;t); (2.8) where xt;t for  = 1 is understood to be xt;t−1, the one-period-ahead forecast of the output gap in period t − 1. Thus, the optimal targeting rule in this example can be expressed as “@nd an instrument-rate path so the in ation-gap forecast is −=x times the change in the output-gap forecast”. 6 In this example, optimal “in ation-forecast targeting” can then be described as fol￾lows: (1) Conditional on the judgment zt ≡ {zt+;t}∞ =0, @nd in ation and output gap forecasts, t ≡ {t+;t}∞ =1 and xt ≡ {xt+;t}∞ =1, that ful@ll the speci@c target￾ing rule (2.8) and the aggregate-supply relation (2.6). (2) Conditional on the judg￾ment and these forecasts, @nd the instrument-rate forecast, i t ≡ {it+;t}∞ =0, that ful@lls the aggregate-demand relation (2.7). (3) Announce these forecasts and set the instrument-rate accordingly. This results in the optimal instrument-rate setting, condi￾tional on the judgment, zt , without having to specify the optimal reaction function. As is shown in Svensson (2001b), even for this relatively simple model, the optimal reaction function is overwhelmingly complex, especially since it must specify how to respond optimally to judgment, making veri@ability and commitment directly to the optimal reaction function completely unrealistic. 6 As is explained in Svensson (2001b), (2.8) also applies for  = 1, when xt;t is interpreted to be xt;t−1. Formulating the targeting rule this way leads to “optimality in a time-less perspective”, corresponding to a situation of commitment to optimal policy far in the past, as discussed in Woodford (1999) and Svensson and Woodford (1999).