ECON 815. A Basic New Keynesian Model II

Transcription

1 ECON 815 A Basic New Keynesian Model II Winter 2015 Queen s University ECON 815 1

2 Unemployment vs. Inflation Unemployment Core Inflation Unemployment CPI Inflation Figure : Unemployment vs. Inflation Canada Queen s University ECON 815 2

3 Japan Queen s University ECON 815 3

4 Sticky Prices Firms can change their price only with probability (1 θ). Suppose they set their price in period t = 0. In period t = 1, with probability θ they cannot change it. In period t = 2, conditional on not having changed in period 1, they cannot change it with probability θ. And so on... Firms solve: max P t(i) k=0 [ ( θ k E t Q t,t+k subject to ( ) ɛ Pt (i) Y t+k (i) = C t+k P t+k P t (i)y t+k (i) W t+k ( Yt+k (i) A t+k where Q t,t+k captures stochastic equilibrium discounting. ) 1 )] α Queen s University ECON 815 4

5 FOC: k=0 ( θ k E t [Q t,t+k Y t+k (i) P t (i) ɛ )] ɛ 1 ϕ t+k(i) = 0 All firms that can change prices today, will chose the same price, P t. When the price cannot be adjusted in period t + k, the firm sets labour demand to satisfy its demand for goods ( Yt+k (i) N t+k (i) = A ) 1 ( ) α P ɛ = t P t+k α ( C t+k A ) 1 α Hence, firm cannot maintain its desired mark-up µ when its price is sticky. Queen s University ECON 815 5

6 Distortion For fixed P t, we have N t+k (i)/ P t+k > 0. Then, we have that nominal marginal costs ϕ t+k (i) are large. We have that the mark-up is depressed, i.e. µ t < µ, or equivalently that real marginal costs are high. Unexpected price increases will lead to higher labour demand. Hence, mark-ups can be interpreted as a labour wedge that depresses the MPL. Conclusion: Firms would like to increase prices with the price level, but cannot do so. However, they set prices that take into account expected changes in nominal marginal costs. Queen s University ECON 815 6

7 Steady State Suppose we have zero inflation in steady state, Pt there is no need to change prices. = P t 1 so that The following equations describe the steady state Y (i) = Y Y = AN α U n = αa N α 1 U c P 1 = β(1 + ῑ) P = ɛ ɛ 1 ϕ(i) Hence, firms charge identical mark-up and real marginal costs are constant at 1/µ. We now look at equilibrium in a log-linearized version around this steady state. Queen s University ECON 815 7

8 Aggregate Price Level The price index in period t is given by Pt 1 ɛ = P t 1 (i) 1 ɛ di + (1 θ)pt 1 ɛ i fixed The distribution of fixed prices corresponds to the distribution of last periods prices with weight θ, or θpt 1 1 ɛ = P t 1 (i) 1 ɛ di Hence, inflation is given by i fixed [ ( P Π t = θ + (1 θ) t P t 1 ) 1 ɛ ] 1 1 ɛ Inflation changes less than 1-1 with price changes of individual firms. Queen s University ECON 815 8

9 Inflation Dynamics We look at approximations around a zero inflation steady state. The log-linearized firm s FOC and the inflation equation give ( π t = βe t [π t+1 ] + λ log ϕ t log ɛ 1 ) P t ɛ [ = λ β k E t log ϕ t+k log ɛ 1 ] P t+k ɛ where λ = (1 θ)(1 βθ) θ ϕt P t t=0 α α+ɛ(1 α) are average real marginal costs for firms Inflation is given by expected deviations from steady-state mark-up. Inflation is high (low) whenever firms expect real marginal costs above (below) their steady state values. Queen s University ECON 815 9

10 The New Keynesian Philips Curve We can express deviations in real marginal costs in terms of deviations in output. log ϕ t log ɛ 1 ( ) η + (1 α) = σ + (y t yt n ) P t ɛ α The last expression is the output gap which measures the deviation of the actual output level from the (optimal) output level associated with flexible prices, y n t. This links inflation dynamics to dynamics in output, or ( where κ = λ σ + η+(1 α) α π t = βe t [π t+1 ] + κ(y t yt n ) ). Queen s University ECON

11 Summary The NK Trinity 1) Phillips Curve π t = βe t [π t+1 ] + κ(y t y n t ) 2) Intertermporal Euler equation y t y n t = 1 σ (i t E t [π t+1 ] r n t ) + E t [y t+1 y n t+1] where rt n = ρ + σe t [yt+1 n yt n ] is the natural rate of interest which changes due to real (or supply) shocks. 3) Specification of a monetary policy rule. The Phillips Curve specifies inflation in terms of the output gap which is given by the Euler equation through the natural rate and the actual real rate. The latter one is pinned down by monetary policy. Queen s University ECON