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5. the New Neoclassical Synthesis and Monetary Policy Optimising Model with Nominal Rigidities (Dynamic IS AS Model)

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5. the New Neoclassical Synthesis and Monetary Policy Optimising Model with Nominal Rigidities (Dynamic IS AS Model) 5. The New Neoclassical Synthesis and Monetary Policy Optimising Model with Nominal Rigidities (Dynamic IS AS Model) Recent research is focused on dynamic models with micro-foundations Models build a bridge between academics and practitioners (empirically oriented, dynamic structure). Research has strong influence on monetary policymaking. Theory provides different frames and tries to capture stylized facts. Empirical calibrations used to make empirical forecasts. New issues discovered and old results re-emerge in new setting New Neoclassical Synthesis: simple, widely applied small-scale macro model Woodford Chapter 3/ 7; Clarida/Gali/Gertler (1999); Gali (2008, chapter 3) Key ingredients: A) Demand side: dynamic IS-Curve derived from stochastic Euler equation B) Supply side: Phillips-Curve; rigidities reflect real effects of monetary policy. C) Monetary Policy Rule: Derive optimal policy from max welfare 1 A) Demand side: dynamic IS-Curve derived from stochastic Euler equation ˆ n xt Et xt1 [(it Ett1) rˆt ] ˆ ˆ n with output gap: xt Yt Yt (log deviations from steady state) ˆ n Yt : stochastic, hypothetical flex-price output (natural rate of output in a real business cycle model). B) Supply side: Phillips-Curve We consider 3 variants: B1. New Classical Phillips Curve Some prices fixed in advance t const xt Et1 t depends on proportion of firms that prefix prices. B2. New Keynesian Phillips Curve derived from staggered price setting (Calvo price adjustment) t const xt Et t1 depends on proportion of firms that prefix prices. 2 B3. New Keynesian Phillips Curve derived from price setting with sticky information (Mankiw/Reis) j t xt (1 ) Et j1( t xt ) j0 depends on proportion of firms that update information. Price dispersion distorts optimal consumption stabilise price level C) Monetary Policy Rule: Derive optimal policy from max welfare * * * C1. Interest rate rule (Taylor): it it (t ) x (xt x ) C2. Money supply rule: based on equilibrium on money market – requires knowledge about money demand function. ˆ ˆ n C3. Min. welfare losses out of inflation t and output gap xt Yt Yt Second order approximation for losses due to deviations from inflation target * and ˆ ˆ n natural output Yt * Yt : E t ( *)² b (x )² Loss function: min t0 t t s.t. A) and B), eventually second best s.t. C1 or C2. 3 5.1) Derivation of the New Classical Phillips-Curve Some firms fix prices in advance, others respond to current demand. Monopolistic competition á la Blanchard/Kiyotaki. P (i) c (i) const. t Y Demand for good i: t t ; normalize const = 1. Pt Let there be a share (1-α) of firms who set prices in advance: P (i) max E E (P (i) mc ) t1 Y Firms t t1 t t1 t1 t1 Pt1(i) P t1 Y Y P (i) E t1 E (P (i) mc ) t1 P (i) 1 0 FOC: t1 t t t1 t1 t1 Pt1 Pt1 Y Y P (i)1 E t1 E mc t1 0 t1 t t t1 Pt1 Pt1 4 Yt1 Yt1 Pt1 (i)1 Et Et mct1 0 Pt1 Pt1 Yt1 Et mct1 Pt1 * Pt1(i) (1 )Et mct1 Et Pt1 Yt1 1 , Et Pt1 1 where we assume that fluctuations of output and price level are small. (Note that 1 ). If only a share α of firms adjusts its price to current demand, we get P P* E P* * * t 1 t 1 (1) t t 1 Pt 1 Pt 1 (1)Et Pt 1 <=> Pt Pt Pt * * <=> 1 t 1 (1 t 1) (1)(1 Ett 1) * * <=> t 1 t 1 (1)Et t 1 * * log-approximation: pt 1 pt ( pt 1 pt ) (1)(Et pt 1 pt ) * * <=> pt 1 pt 1 (1)Et pt 1 5 The aggregate price level (in logarithmic terms, approximation) is * * * * * pt pt (1)Et 1 pt . Note that Et1 pt Et1 pt (1)Et1 pt Et1 pt * Hence, (1) pt (1)Et1 pt ( pt pt ) p ( p* p ) E p P* (1 )MC t 1 t t t1 t . Using t t , (ln(1 ) ln mc p ) E p 1 t t t1 t (ln(1 ) ln(rmc )) E p 1 t t 1 t , rmc = real marginal costs. If labor supply is increasing in the real wage, log real marginal cost rise with the output gap. Assuming a linear relation between log real marginal cost and output gap ln(rmct ) xt , we get the New Classical Phillips curve Pt t const. xt Et 1 t , where we use t 1 ln(Pt / Pt1 ) pt pt1 . Pt1 Result: Current output depends on current inflation and past expectation of current inflation. 6 5.2) Derivation of the Forward looking Phillips-Curve Based on staggered price setting process by Calvo. Very stylised approach to represent sticky prices, designed to ensure simple aggregation, independent of the history Intuition: Stochastic price adjustment: in each period any firm faces a state-independent probability of being stuck with its old price. is independent of when the firm last changed its price 1 Probability that a firm gets the chance to revise it’s price, independent of other firms and of its own history (needed for stationary process) introduces forward looking component t Probability that firm’s price cannot be changed until period t Expected duration of price stickiness (expected length of fix prices): 1 2 3 ... 1/(1) (Poisson process) * Let pt be the optimal price of a firm in period t. Assuming symmetry it is the same for all firms who can adjust their price in t. 7 The price level evolves according to * (1) pt pt1 (1) pt * Partial price adjustment towards optimum: pt pt1 (1) [pt pt1] * Rational expectations: pt is the optimal price for a firm who expects that its price may be fixed for several periods. Firms must be forward looking! * Determine pt : Forward looking price setting: p (i) c (i) const. t Y Demand for good i: t t pt pt (i) max Et ( ) t Et ( ) ( pt (i) mct ) Yt Firms p (i) t 0 0 pt Y E ( ) t p (i) ( p (i) mc ) p (i) 1 0 FOC: t t t t t 0 pt Y E ( ) t p (i)(1 ) mc ) 0 t t t 0 pt 8 Y ( ) E t mc t t 0 p p (i) t (1 ) E ( mc ) t t t Y 1 0 ( ) E t t 0 pt Assuming small fluctuations in output and price level, Et mct Et mct (1 ) Et mct . 0 Define (1 ) , so that coefficients are summing up to 1. Adjusting firms set their prices as a mark-up over a weighted average of current and expected future nominal marginal costs mc t (apart from cost push shocks u ). Given these shocks, p* (1 )(1 ) ( ) E (mc ) u t 0 t t t . (*) 9 p* (1 )(1 ) ( ) E (mc ) u t 0 t t t (*) p* (1 )(1 ) ( ) 1 E (mc ) u => t1 1 t1 t t1 E p* (1 )(1 ) ( ) E (mc ) <=> t t1 1 t t Replacing the sum in (*) for tau=1 to infinity by the previous equation, * * (2) pt (1 )(1 )mct Et ( pt1) ut . * recall (1): pt pt1 (1) pt Recursive form (stationary structure as a result of Poisson process): - (1) Actual price is a function of optimal price today and the price last period - (2) Optimal Price is a function of current marginal cost and expected optimal price tomorrow * (1) holds also for 1, pt 1 pt (1) pt1 multiply by and subtract from (1) to get: 10 * 2 * pt Et pt 1 pt1 (1) pt pt (1) Et pt1 * * ( pt 1 pt ) (1) ( pt Et pt1 ) Using (2) we get: pt Et pt1 ( pt1 pt ) (1)(1 )(1 ) mct (1) ut ( pt pt 1) 2 Et pt1 (1 ) pt (1)(1 )(1 ) mct (1) ut pt pt1 1 1 1 E p ( ) p ( )(1 )(1 ) mc ( ) u t t1 t t t ( pt pt 1)/ pt p p 1 1 t 1 t ~ Et ( ) ( )(1 )(1 ) rmct ut pt 1 const E (1 )(1 ) rmc u~ t t t 1 t t 11 If we assume that real marginal cost rise linear with the output gap rmct xt , we get as aggregate supply curve (forward looking Phillips curve) ~ t const xt Et t1 ut 1 ˆ ˆ n (1 )(1 ) with output gap: xt Yt Yt and κ is an (inverse) measure of nominal rigidity: The higher α, the lower κ (for α 0, κ ∞) ~ ut : captures variation in prices not related to output gap (e.g. fluctuations in firm’s mark up; cost push shocks) Transmission mechanism for monetary policy: Consider expansionary monetary policy. Output is determined by demand, prices are sticky. Firms who cannot adjust their prices face higher demand and, thus, produce more (real effect). Higher demand for labor increases real wage due to convexity of disutility from work. Firms who can adjust prices, will raise prices (1) in order to meet rising demand in t, and (2) in response to expected rise of prices of other firms in the future. Prices are strategic complements! Higher expected prices of competitors in the future provide an incentive to raise own price today if future price adjustments are possible only with a probability below 1. forward looking behaviour. 12 Key results: Price stickiness, but no stickiness in inflation. Adjusting firms raise prices more than would be optimal if they could be sure to adjust again in the future. No inflationary inertia (contrary to empirical results). Fairly robust result, does not depend on specific Calvo price setting mechanism. Disinflation can be achieved without cost, even have positive effects (see: Fuhrer/Moore): An announced contractionary monetary policy leads firms to reduce prices already today.
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