Macroeconomics Basic New Keynesian Model Nicola Viegi April 29, 2014
The Problem I Short run E ects of Monetary Policy Shocks I I I persistent e ects on real variables slow adjustment of aggregate price level liquidity e ect I Micro Evidence on Price-setting Behaviour: signi cant price and wage rigidities I Failure of Classical Monetary Models
The Evidence
The Evidence
The Evidence: South Africa
Why New Keynesian? I Real Rigidities - Monopolistic Competition I Nominal Rigidities - Calvo Pricing I But with Microfundations I I Method RBC literature (Dynamic Stochastic General Equlibrium) Explain Persistence and Demand Shocks
The Simple Macro Structure Supply (NK Phillips Curve) π t = βe t π t+1 + αy t + ε t (1) Demand (NK IS) y t = E t y t+1 γ (i t E t π t+1 ) + η t (2) Monetary Policy Rule (Taylor Rule) i t = r t + φ π (π t π ) + φ y (y t ) (3)
Where Do They Come From? π t = βe t π t+1 + αy t + ε t (4) Derived from optimal pricing behaviour of the rm with market power and nominal stickiness (Calvo Pricing) y t = E t y t+1 γ (i t E t π t+1 ) + η t (5) Directly from the rst order condition of the consumer and equilibrium conditions Monetary policy reaction function Note: i t = r t + φ π (π t π ) + φ y (y t ) (6) I There is no need of having an LM (no explicit money) I Both π t and y t are jumping variables (function of expectations of future state variables)
A Preliminary Issue: CES Aggregator In modern macro models with imperfect competition, the Dixit-Stiglitz, or Constant-Elasticity-of-Substitution aggregator plays an important role. Consider a static optimization problem of a rm that buy in nite number of intermediate products C (i), puts them together using technology Z 1 C = 0 2 C (i) 2 2 1 2 1 di (7) where 2 is the elasticity of substitution (and also the price elasticity of the demand function). Its optimization problem is Z 1 max C P C (i)p(i)di C (i) 0 subject to the production technology above
CES Aggregator The optimality conditions are given by P(i) 2 C (i) = C (8) P This is also the demand function of a good C (i). (You must work out the details by yourself.) Plug this to the pro ts and use zero pro t constraint to get the aggregate price level (=price index=marginal costs): Z 1 P = P(i) 1 0 1 ɛ 2 1 di (9)
Consumer Problem E t i=0 " C 1 σ β i t+i 1 σ # χ N1+η t+i 1 + η (10) Z 1 C t = 0 θ c θ θ 1 θ 1 jt dj (11) Z 1 P t = 0 pjt 1 θ 1 1 θ dj (12) P t C t + B t = W t N t + (1 + r t ) B t 1 + Π t (13)
Consumer Problem Optimal Allocation of Consumption Expenditure subject to Z 1 min c ij 0 p jt c jt dj (14) Optimal Allocation Z 1 C t = 0 θ c θ θ 1 θ 1 jt dj (15) c jt = pjt P t θ C t (16)
Consumer Problem Optimal Dynamic Consumption/Leisure Decision E t i=0 " C 1 σ β i t+i 1 σ # χ N1+η t+i 1 + η (17) Lagrangian C t + B t = W t N t + (1 + r t ) B t 1 + Π t (18) P t P t P t P t L = E t i=0 E t i=0 " C 1 σ β i t+i 1 σ # χ N1+η t+i + (19) 1 + η # " Wt+i β i P λ t+i N t+i + (1 + r t+i ) B t+i 1 P t+i t+i + Π t+i P t+i C t+i B t+i P t+i (20)
Consumer Problem Optimal Dynamic Consumption/Leisure Decision FOC L = Ct σ + λ t+i = 0 (21) C t L = χn η W t t + λ t = 0 (22) N t P t L 1 1 = λ t + E t βλ t+1 (1 + r t+1 ) = 0 (23) B t P t P t+i
Consumer Problem Optimal Dynamic Consumption/Leisure Decision Rearranging and using condition (21) to eliminate the Lagrange multiplier, we get: χn η t = Ct σ W t C σ t = E t β (1 + r t+1 ) (24) P t Pt Ct+1 σ (25) P t+i As shown before, this implies the following log linear relationships w t p t = σc t + ηn t (26) 1 c t = E t fc t+1 g σ fi t E t π t=1 ln βg (27)
Firm Problem Calvo Pricing I A rm may change price of its product only when Calvo Fairy visits. I The probability of a visit is (1 θ) I It is independent of the length of the time and the time elapsed since the last adjustment. Hence, in each period the (1 θ) share of rms I may change their price and rest, θ, keep their price unchanged. I Mathematically, Calvo Fairy s visits follows Bernoulli process (discrete version of Poisson process). I The probability distribution of the number of periods between the visits of Calvo Fairy is geometric distribution. I The expected value of geometric distribution and, hence, the average number of periods between the price changes (of a rm) is 1 1 θ
Firm Problem Calvo Pricing Price Stickiness: Firms can change prices with a probability (1 θ) Price Decision - Intertemporal Problem of the Firm (when they can change the prices) min L (z t ) = z t (θβ) k E t (z t pt+k ) 2 (28) k=0
FOC L 0 (z t ) = 2 " k=0 (θβ) k E t (z t p t+k ) = 0 (29) # (θβ) k z t = (θβ) k E t (pt+k ) (30) k=0 k=0 z t 1 θβ = (θβ) k E t (pt+k ) (31) k=0 z t = (1 θβ) k=0 (θβ) k E t (p t+k ) (32)
Reset Price z t = (1 θβ) p t + θβe t z t+1 (33) where E t z t+1 = frictionless optimal price p. (θβ) k E t (pt+1+k ) k=0 Thus the reset price can be written as: p t = µ + mc t (34) z t = (1 θβ) (µ + mc t+k ) + (1 θβ) E t z t+1 (35)
Aggregate Pricing This can be rearranged as p t = θp t 1 + (1 θ) z t (36) z t = 1 1 θ (p t θp t 1 ) (37) 1 1 θ (p t θp t 1 ) = θβ 1 θ (p t+1 θp t ) + (1 θβ) (µ + mc t ) this equation implies: π t = βπ t+1 + 1 θ θ (1 θβ) (µ + mc t p t ) (38)
New Keynesian Phillips Curve Denoting cmc t = µ + mc t p t π t = βπ t+1 + 1 θ θ (1 θβ) cmc t (39) cmc t = λy t π t = βπ t+1 + γy t (40) where γ = λ (1 θ) (1 θβ) θ
In ation as an Asset Price Soving Forward the New Keynesian Phillips curve (as for the Cagan Model) π t = γ k=0 β k E t y t+k Problems I Not enough dynamic in the model I Any change will be re ected instantaneously on the variables I Many attempts to produce more slugghish response (will see later)