Jordi Gali Monetary Policy, inflation, and the business cycle Lian Allub 15/12/2009
In The Classical Monetary economy we have perfect competition and fully flexible prices in all markets. Here there is a very limited roled to money (serve as a unit of account). Main conclusion: Real variables are determined independently of monetary policy. Monetary policy is neutral As utility is a function of consumption and hours work then there is no policy rule better than other. Monetary policy shocks affect nominal variables but the way that it affects has conflict with the empirics. Also cannot predicts the effects on real variables. Those empirical failures are the main motivation behind the introduction of nominal frictions.
The 2 key elements of the Basic New Keynesian Model (NKM) are: Imperfect Competition in the goods market. Each Firm produces a differentiated good for which it sets the price. Only a fraction of firms can reset their prices in any given period
Households Reppresentative infintely-lived household. max Ct,N t E 0 t=0 β t U(C t, N t ) st 1 0 P t(i)c t (i)di + Q t B t B t 1 + W t N t + T t ; t = 0, 1, 2,... ( ) 1 where C t 0 C t(i) 1 1 ɛ ɛ 1 ɛ di Assume there is a continuum of goods in [0,1]. B t one-period, nominal riskless discount bonds purchased in t and maturing in t + 1. Q t (i) is the price of the bond. T t are lump-sums additions or subtractions in nominal terms. Solvency Condition lim T E t {B T } 0 for all t
The household must decide on the consumption, labor and how to consume each variety. For the latter solves max Ct (i) C t st 1 0 P t(i)c t (i)di = Z t The solution to this is: ( ) Pt (i) ɛ (1) C t (i) = C t where P t [ P t Conditional on this behavior we can write 1 0 P t(i)c t (i)di = P t C t Optimality Condition Q t = βe t { Uc,t+1 U c,t U n,t U c,t = W t P t P t+1 P t } 1 0 P t (i) 1 ɛ di] 1 1 ɛ
Using a utility function of the form U(C t, N t ) = C t 1 σ we can write the optimality conditions in log terms as: 1 σ N1+ϕ t 1+ϕ (2) (3) w t p t = σc t + ϕn t c t = E t {c t+1 } 1 σ (i t E t {π t+1 } ρ) where i t logq t and ρ log β When necessary we will use an ad-hoc money demand function: (4) m t p t = y t ηi t
Firms Assume a continuum of firms i [0, 1]. Each firms produced a differentiated good, but they use the same technology. (5) Y t (i) = A t N t (i) 1 α where A t represent the level of technology common to all firms and evolves exogenously over time. Each firms face demand schedule given by (1) and take P t and C t as given. In each period a measure 1 θ of firms reset their prices while a fraction θ keep their prices unchanged. So the average duration of a price is given by (1 θ) 1 and θ becomes an index of price stickiness.
Aggregate Price Dynamics Aggregate price dynamics are described by the equation: ( P (6) Πt 1 ɛ ) 1 ɛ = θ + (1 θ) t Pt 1 where Π t P t P t 1 and P t is the price set in period t by firms reoptimizing their price in that period In steady state Π = 1 because P t = P t 1 = P t. A log linear approximation of (6) around the ss gives: (7) π t = (1 θ)(p t p t 1 ) Inflation comes from the fact that firms reoptimizing prices today set prices different than the economy s average price in the previous period.
Optimal Price Setting A firm reoptimizing in t will maximize the current market value of the profits generated while the price chosen is effective. For that solves: max P t k=0 } θ k E t {Q t,t+k (Pt Y t+k t Ψ t+k (Y t+k t )) (8) ( P ) ɛ sty t+k t = t C P t+k for k = 0, 1, 2... t+k where Q t,t+k β k (C t+k /C t ) σ (P t /P t+k ) is the stochastic discount factor for nominal payoffs, Ψ(.) is the cost function, and Y t+k t output in period t + k for a firm that reset price in period t
The FOC of this problem (dividing by P t 1 ) gives: (9) { ( θ k P )} E t Q t,t+k Y t t+k t MMC P t+k t Π t 1,t+k = 0 k=0 t 1 where M ɛ 1 ɛ, MC t+k t ψ t+k t /P t+k (real marginal cost in t + k for firm whose price was last reset in t) and Π P t+k /P t If θ = 0 then P t = Mψ t t which is the opitmal price-setting condition under flexible prices. M desired Mark-up A first order aproximation of the FOC around the zero inflation steady state yields: (10) pt { } = µ + (1 βθ) (βθ) k E t mc t+k t + p t+k k=0 where mc = µ and µ logm The price choosen by firms is the mark up plus a weighted average of their current and expected marginal cost.
Equilibrium Let Y t ( 1 0 Y t(i) 1 1 ɛ di) ɛ 1 ɛ. So the goods Market clearing is Y t (i) = C t (i) Y t = C t, which combines with the Euler eq. yield the eqm. condition: (11) y t = E t {y t+1 } 1 σ (i t E t {π t+1 } ρ) Labor market clearing: N t = 1 0 N t(i)di which together with (5) and taking logs gives: (1 α)n t = y t a t + d t where d t (1 α)log 1 0 (P t(i)/p t ) 1 α ɛ Around the ss of zero inflation d t = 0 so: y t = a t + (1 α)n t The economy s marginal cost is (using also the last expression) mc t = (w t p t ) 1 1 α (a t αy t ) log(1 α)
Using this equation with the market clearing condition and the demand schedule (1): (12) mc t+k t = mc t+k αɛ 1 α (p t p t+k ) Now using (12) and (10) and rearringing terms we get: (13) pt p t 1 = βθe t {pt+1 p t } + (1 βθ)θmc ˆ t + π t where θ 1 α 1 α+αɛ 1, and mc ˆ t mc t mc is the log deviation of mg cost from its ss value. Using (7) and (13) yields the inflation equation (14) π t = βe t {π t+1 } + λmc ˆ t where λ (1 θ)(1 βθ) θ Θ is strictly decreasing in θ, α, andɛ
Solving forward (14) we can write π t = λ k=0 βk E t { mc ˆ t+k }. Inflation will be high when firms expect average mark ups to be below their ss level (µ) for in that periods firms reseting prices will set a higher price than the economy s average price. Relation between the economy s real Mg cost and a measure of aggregate economic ativity. ( ) (15) mc t = σ + ϕ+α 1 α ( (16) mc = σ + ϕ+α 1 α (17) yt n = ψyaa n t + ϑy n where ϑ n y (1 α)(µ log(1 α)) σ(1 α)+ϕ+α y t 1+ϕ 1 α a t log(1 α) ) y n t 1+ϕ 1 α a t log(1 α) > 0 and ψ n ya 1+ϕ σ(1 α)+ϕ+α Subtracting (16) from (15) and making (y t yt n ) y t ( (18) mc ˆ t = σ + ϕ + α ) y t 1 α
Using (18) and (14) we can get the New Keynesian Phillips Curve (19) π t = βe t {π t+1 } + κy t where κ λ(σ + ϕ + α 1 α ) Rewriting (11) we can get the Dynamic IS equation (DIS) (20) (21) y t = 1 σ (i t E t {π t+1 } rt n ) + E t { y t+1 } rt n { } ρ + σe t y n t+1 r n t = ρ + σψ n yae t { a n t+1 } where r n t is the natural interest rate. Assuming lim T E t { y t+t } = 0 we can write (22) as: (22) y t = 1 σ (r t+k rt n ) where r t i t E t {π t+1 } k=0 In order to close the model we need to specify how i evolves over time i.e how the monetary policy is conducted
Interest Rate Rule We have an interest rate rule of the form: (23) i t = ρ + φ π π t + φ y y t + v t, v t is exogenous and 0 mean Equilibrium conditions are represented by: [ ] [ ] y t E t { y t+1 } ( ) = A T + B T r ˆ t n v t E t {π t+1 } π t We will use the following values for the exercises β = 0.99, σ = 1, φ = 1, α = 1/3, ɛ = 6, η = 4, θ = 2/3, φ pi = 1.5, φ y = 0.5/4, ρ v = 0.5
A monetary policy shock The effect of a monetary shock. Assume v t = ρ v v t 1 + ɛ v t. Take an increase in ɛ v t of 25 basis points. The results of this policy are: 1. An increase in the real rate 2. a decrease in output and inflation. 3. Nominal rate goes up but less than the real rate 4. Reduction in the money supply. We have a liquidity effect
Technology shock. Assume a t = ρ a a t 1 + ɛ a t where ɛ a t is a zero mean white noise process. In general the sign of the response of output and employment to a positive technology shock is ambiguous. In the exercise, a positive technology shock gives: 1. a decline in the nominal and real interest rate and increase in quantity of money. 2. Negative output gap (the output increase is less than the natural output increase) and a decline in inflation. 3. Employment decline
Exogenous Money Supply ( m) The Equilibrium Dynamics can be summarized in the following system: A M,0 y t π t l t 1 E t { y t+1 } = A M,1 E t {π t+1 } lt l t m t p t and l t 1 = l t + π t m t + B M ˆ r n t y n t m t
The effect of a monetary Policy shock m t = ρ m m t 1 + ɛ m t where ɛ m t is WN We analyze the effect of a shock to ɛt m = 0.25 while setting rˆ t n = yt n = 0 for all t. Assume ρ m = 0.5 In our calibration example we have: 1. Output increases first and then converges to it s original level 2. Nominal rate shows a slight increase 3. Real rate declines persistently (due to the higher expected inflation) expansion in output and aggregate demand persistent rise in inflation
A technology Shock Assume a t = ρ a a t 1 + ɛ a t where ɛ a t is a zero mean white noise process. We set m t = 0 The results of a positive shock are: 1. We have a negative output gap (as in the previous case) and hence a decline in inflation. In this case the gap is much larger, so we also have a larger decline in employment 2. Money supply does not change real rate increases. Contrasts with the previous case