Consider the following basic New Keyne- sian model of a closed economy. -0. Rt - Et (Tl++1)] + E+ (x++1) + dt, o > 0, (Q12.17) att Ax+ +BE+ (T4+1) + St, > 0, B > 0, (Q12.18) de Podt-1 + εt, 0 <PD < 1, (Q12.19) St. PsSt-1 + nt, 0 <ps <1, (Q12.20) where x+ = yt – Zt is the output gap (the difference between actual stochastic output, yt, and the natural output level, zt), Rt is the nominal interest rate, T = Pt/Pt-1 - 1 is the inflation rate (Pt being the price level), d? is a stochastic demand shock, and St is a stochastic supply shock. Except for R4 and Tlt, all variables are measured in logarithms. The innovation terms, Et and 1t, are independent and identically distrib- uted random variables with mean zero (E+ (8t) = Et (9t) = 0) and constant variances (E+ (87) = o? and E4 (17) = 0%)). Equation (Q12.17) is a New Keynesian “IS curve" relating the current output gap negatively to the real interest rate, and positively to the expected future output gap and the demand shock. Equation (Q12.18) is a New Keynesian "Phillips curve” relating the inflation rate to the output gap, expected fu- ture inflation, and the supply shock. The nominal interest rate is assumed to be the instrument of monetary policy. The private sector is blessed with rational expecta- tions. The central bank has the following objective function which it wants to minimize: = = 1 Ω, = 2 Et Epi. [ax?+i+72+1] . (Q12.21) i=0 where a > 0 is the relative weight placed on output gap fluctuations. (a) Show that in this model the current output gap depends not only on the current real interest rate and demand shock, but also on the expected value of future real interest rates and demand shocks. (b) Show how the New Keynesian Phillips curve differs from the standard ex- pectation augmented Phillips curve. Show that in the New Keynesian Phillips curve there is no lagged dependence in inflation.

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