New Keynesian Model Walsh Chapter 8

New Keynesian Model Walsh Chapter 8

1 General Assumptions Ignore variations in the capital stock There are differentiated goods with Calvo price stickiness Wages are not sticky Monetary policy is a choice of the nominal interest rate

2 Households Maximize expected utility which depends on a composite consumption good, real money, and leisure β i C1 σ t+i + γ ( ) 1 b Mt+i χ N 1+η t+i 1 σ 1 b 1 + η E t i=0 Composite consumption good is CES P t+i C t = [ 1 θ 1 c θ jt dj 0 ] θ θ 1 θ > 1

Choose c jt to minimize the cost of buying C t subject to [ 1 1 min c p jt jtc jt dj 0 θ 1 c θ jt dj 0 ] θ θ 1 Ct set up a Lagrangian L = 1 0 p jtc jt dj + Ψ t C t ( 1 θ 1 c θ jt dj 0 ) θ θ 1

First order condition with respect to c jt θ p jt Ψ t θ 1 ( 1 θ 1 c θ jt dj 0 ) θ θ 1 1 θ 1 θ 1 c θ jt = 0 p jt Ψ t ( 1 θ 1 c θ jt 0 p jt Ψ t C dj ) 1 θ 1 c 1 θ jt = 0 1 θ t c 1 θ jt = 0 c jt = ( ) θ pjt C t Ψ t

Integrate over j to solve for C t and eliminate the multiplier C t = 1 pjt ( 0 Ψ t ) θ C t dj θ 1 θ θ θ 1 = Ψ θ t C t [ 1 0 p1 θ jt dj ] θ θ 1 Solve for Ψ t 1 = Ψ t [ 1 0 p1 θ jt dj ] 1 θ 1 Ψ t = [ ] 1 1 1 θ 0 p1 θ jt dj Pt

Substituting into demand yields demand as a function of relative price and of composite consumption c jt = ( ) θ pjt C t P t the price elasticity of demand is θ > 1 Budget constraint for household C t + M t + B t = W t N t + M t 1 P t P t P t P t where Π t is firm profits + (1 + i t 1) B t 1 P t + Π t

Household optimization problem define ω t = (m t 1 + (1 + i t 1 ) b t 1 ) 1 + π t = C t + m t + b t w t N t Π t substitute b t out by using b t = ω t C t m t + w t N t Π t value function V (ω t ) = max ( C 1 σ ) t 1 σ + γ 1 b ( Mt P t ) 1 b χ N t 1+η +βe t V (ω t+1 ) 1 + η ω t+1 = (m t + (1 + i t ) (ω t C t m t + w t N t Π t )) 1 + π t+1

first order conditions using envelope condition C σ t βe t V ω (ω t+1 ) (1 + i t) 1 + π t+1 = 0 γm b t βe t V ω (ω t+1 ) i t 1 + π t+1 = 0 χnt η + βe 1 + i t tv ω (ω t+1 ) w t = 0 1 + π t+1 V ω (ω t ) = βe t V ω (ω t+1 ) 1 + i t = Ct σ 1 + π t+1

FO conditions become C σ t = βe t C σ t+1 (1 + i t ) 1 + π t+1 γm b t = βe t C σ t+1 i t i t = Ct σ 1 + π t+1 1 + i t χnt η = βe tct+1 σ w t (1 + i t ) = Ct σ 1 + π t+1 w t

Firms maximize profits subject to constraints Constraints production function c jt = Z t N jt EZ t = 1 demand curve c jt = ( ) θ pjt C t P t firm can adjust price with probability 1 ω choose labor to minimize costs taking the real wage as given min w t N jt + ϕ t ( cjt Z t N jt )

first order condition ϕ t = w t Z t = w t MP N = real marginal cost choose price ( p jt ) to maximize real discounted profits E t i=0 where discount factor is [ ] ω i pjt i,t+i c jt+i ϕ t+i c jt+i P t+i i,t+i = β i ( Ct+i C t ) σ

since c jt depends on price through demand, substitute demand curve for c jt E t i=0 ω i i,t+i ( pjt P t+i ) 1 θ ϕ t+i ( ) θ pjt C t+i P t+i optimal p jt = p t since firms are identical in all ways except date at which last changed price: derivative with respect to p jt E t i=0 ω i i,t+i ( ) θ pjt (1 θ) + ϕ t+iθ P t+i ( pjt P t+i ) θ 1 C t+i P t+i = 0 E t i=0 ( ) ω i p i,t+i [(1 θ) t P t+i + ϕ t+i θ ] ( 1 p t ) ( p t P t+i ) θ C t+i = 0

first term E t i=0 ω i i,t+i (1 θ) p t P t+i P θ t+i C t+i = p t P t (1 θ) E t i=0 ω i i,t+i P t P θ 1 t+i C t+i second term E t i=0 ω i i,t+i ϕ t+i θp θ t+i C t+i = θe t i=0 ω i i,t+i ϕ t+i P θ t+i C t+i

solve for relative price p t P t = = ( θ θ 1 ( θ θ 1 ) Et i=0 ω i i,t+i ϕ t+i P θ t+i C t+i E t i=0 ω i i,t+i P t P θ 1 t+i C t+i ) E t i=0 ω i i,t+i ϕ t+i ( Pt+i P t ) θ Ct+i E t i=0 ω i i,t+i ( Pt+i P t ) θ 1 Ct+i where last equality multiplies numerator and denominator by P θ t use p t P t = ( θ θ 1 i,t+i C t+i = β i C 1 σ t+i Cσ t ) E i=0 t ω i β i ( ) Pt+i θ ϕ t+i P C 1 σ t t+i E i=0 t ω i β i ( ) P t+i θ 1 P C 1 σ t t+i

Flexible price equilibrium every firm adjusts every period, so ω = 0; lose all but first term p ( ) t θ = ϕ t = µϕ t P t θ 1 implying that price is a markup over marginal costs since θ > 1 since price exceeds marginal cost, output is ineffi ciently low since all firms charge the same price ϕ = 1 µ firms choose labor such that Z t µ = w t

households choose labor such that χn η t C σ t = w in flexible price equilibrium C t = Y t = Z t N t = ( 1 χµ ) 1 σ+η Z 1+η σ+η t Full employment output is potentially affected by shocks to productivity (Z t ) tastes (χ) demand elasticity (markup) (µ)

Aggregate price level recall [ ] 1 1 1 θ 0 p1 θ jt dj Pt [ ] 1 0 p1 θ jt dj = P 1 θ t Aggregate price level P 1 θ t = (1 ω) (p t ) 1 θ + ωp 1 θ t 1 1 ω firms adjust this period and charge the optimal price ω do not adjust, and since the adjusting firms are drawn randomly, the price level for non-adjusters is unchanged

3 Linearized New Keynesian Model New Keynesian Phillips Curve p t P t = ( θ θ 1 ) E i=0 t ω i β i ( ) Pt+i θ ϕ t+i P C 1 σ t t+i E i=0 t ω i β i ( ) P t+i θ 1 P C 1 σ t t+i P 1 θ t = (1 ω) (p t ) 1 θ + ωp 1 θ t 1 Let the relative price the firm chooses when he adjusts be Q t = p t P t Q = 1 in steady state and when all firms can adjust every period

dividing second equation by P 1 θ t 1 = (1 ω) Q 1 θ t + ω ( ) 1 θ Pt 1 P t expressed in percent deviations about steady state with P t 1 P t = 1 1 = (1 ω) (1 + (1 θ) ˆq t ) + ω (1 (1 θ) ˆπ t ) ˆq t = ω 1 ω ˆπ t rewrite first equation as Q t E t i=0 ( ) θ 1 ω i β i Pt+i C 1 σ ( ) θ P t+i = µe t ω i β i Pt+i ϕ t+i C t i=0 P t+i 1 σ t

Approximate the left hand side as C 1 σ 1 ωβ + +C 1 σ i=0 ( C 1 σ 1 ωβ Approximate the right hand side as µ ( C 1 σ 1 ωβ ) ϕ ) ˆq t ω i β i [ (1 σ) E t Ĉ t+1 + (θ 1) (E tˆp t+i ˆp t ) ] +µϕc 1 σ ω i β i [ ] (1 σ) E t Ĉ t+1 + θ (E tˆp t+i ˆp t ) + E tˆϕ t+i i=0

Equating and noting that µϕ = 1 ( 1 1 ωβ ( 1 ) 1 ωβ ˆq t + ) i=0 ˆq t = ω i β i [( 1) (E tˆp t+i ˆp t )] = i=0 ˆp t = ˆq t + ˆp t = (1 ωβ) i=0 ω i β i [ E t (ˆϕt+i + ˆp t+i )] ( i=0 ω i β i [ E tˆϕ t+i ] 1 1 ωβ ω i β i [ E t (ˆϕt+i + ˆp t+i )] the optimal nominal price equals the expected discounted value of future nominal marginal costs can write equation as ωβe t (ˆq t+1 + ˆp t+1 ) = ˆq t + ˆp t (1 ωβ) (ˆϕ t + ˆp t ) ) ˆp t

Solving for ˆq t ˆq t = (1 ωβ) ˆϕ t + ωβ [E t (ˆq t+1 + ˆp t+1 ) ˆp t ] ˆq t = (1 ωβ) ˆϕ t + ωβe t (ˆq t+1 + π t+1 ) using ˆq t = ω 1 ω ˆπ t to eliminate ˆq t ω 1 ω π t = (1 ωβ) ˆϕ t + ωβe t ( ) ω 1 ω π t+1 + π t+1 π t = (1 ωβ) (1 ω) ˆϕ t ω + βe t π t+1 π t = κˆϕ t + βe t π t+1 Differences between the New Keynesian and old Keynesian Phillips Curves

no backward-looking terms, expected future inflation matters, not lagged inflation marginal cost instead of output gap under some restrictions the same from household s labor supply decision, real wage must equal marginal rate of substitution between leisure and consumption using ŵ t ˆp t = ηˆn t + σŷ t ĉ t = ŷ t = ˆn t + ẑ t

marginal costs equals real wage divided by marginal product of labor (Z t ) where implying that ˆϕ t = ŵ t ˆp t ẑ t = ŵ t ˆp t (ŷ t ˆn t ) = ηˆn t + σŷ t ẑ t = η (ŷ t ẑ t ) + σŷ t ẑ [ t = (η + σ) ŷ t 1 + η ] (η + σ)ẑt ŷ f t = 1 + η ˆϕ t = (η + σ) [ ŷ t ŷ f t (η + σ)ẑt New Keynesian Phillips Curve becomes ] = γ [ŷt ŷ f t π t = κx t + βe t π t+1 ]

more complicated when do not have constant returns to scale, but principle is same IS curve ŷ t = E t ŷ t+1 1 σ (î t E t π t+1 ) expressed in terms of the output gap x t = ŷ t ŷ f t x t = E t x t+1 1 σ (î t E t π t+1 ) + u t where u t = E t ŷ f t+1 ŷf t Taylor Rule for nominal interest rate î t = δ π π t + δ x x t + v t

4 Uniqueness of the equilibrium Substitute for the interest rate to get a two-equation system in two unknowns E t π t+1 = 1 β [π t κx t ] E t x t+1 = x t + 1 σ [δ ππ t + δ x x t + v t ] u t 1 σβ [π t κx t ] In matrix form [ Et π t+1 E t x t+1 ] = 1 β βδ π 1 σβ κ β 1 + βδ x+κ βσ [ πt x t ] + [ 0 v t σ u t ]

For uniqueness of equilibrium, since there are two forward-looking variables, the roots of the characteristic equation must both be greater than one equivalently, the system must be unstable letting roots be θ, characteristic equation [ ] [ 1 β θ 1 + βδ ] x + κ θ + κ βσ β [ βδπ 1 σβ ] = 0

Stability using phase diagram Set expected changes to zero (strictly correct if shocks permanent) E t π t+1 π t = 1 β β π t κ β x t = 0 E t x t+1 x t = βδ π 1 σβ π t + βδ x + κ x t + v t βσ σ u t = 0 along E t π t+1 π t = 0 π t = κ 1 β x t slope of π = 0 is positive along E t x t+1 x t = 0 π t = βδ x + κ 1 βδ π x t + β 1 βδ π (v t σu t ) slope of x = 0 depends on sign of 1 βδ π

Case 1: 1 βδ π < 0 slope of x = 0 is negative, implying that slope of π = 0 > slope of x = 0 κ 1 β > βδ x + κ 1 βδ π κ (1 βδ π ) < (βδ x + κ) (1 β) κβ (δ π 1) βδ x (1 β) < 0 κ (δ π 1) + δ x (1 β) > 0 (1) model is globally unstable, equivalently both roots are outside the unit circle unique equilibrium at intersection and otherwise explosion

Case 2: 1 βδ π > 0 and slope of π = 0 > slope of x = 0 sign of (1) is reversed model is saddlepath stable can begin anywhere along the saddlepath and model approaches long-run equilibrium Case 3: 1 βδ π > 0 and slope of π = 0 < slope of x = 0 equation (1) holds (begin with sign reversed and do not flip on second line) model is globally unstable unique equilibrium at intersection

Equation (1) is necessary and suffi cient for global instability and unique equilibrium uniqueness of equilibrium requires a strong enough response of the interest rate to deviations of inflation and output from their steadystate values Positive interest rate shock v t increases Assume 1 βδ π < 0 x = 0 shifts down and inflation and output gap fall as the shock slowly disappears, x = 0 shifts back up and output and inflation return to long-run values

if shock were permanent, output and inflation would never return to steady-state values note output and inflation move together no policy tradeoffs between keeping inflation on target and output on target if no shocks to supply

4.1 Cochrane s Critique 4.1.1 Determinacy How can we rule out all unstable equilibria? Cochrane s answer is that the Fed promises to "blow up the economy" if price and output do not jump to allow non-explosive equilibrium Can the Fed blow up the economy? that is, make price and/or output explode? Output cannot reach ± Price can reach 0 or Interest rate cannot go negative

Cochrane views output trajectory as model flaw should price take off on an unstable trajectory with explosive output, the Calvo fairy would visit more often, as he does in Argentina Yes, the Fed can blow up the economy Would the Fed blow up the economy? Dynamic inconsistency optimal to promise to blow up the economy to prevent it from embarking on an unstable path Not likely to actually carry out promise More likely policy is that economy can get on unstable path and the Fed has plans to move it to a more favorable equilibrium Latter does not rule out the path

4.1.2 New Keynesian Model does not have Old Keynesian dynamics Monetary authority wants to offset positive demand shock Old Keynesian dynamics M i demand with sticky prices Y over time P due to low demand, implying M P back up model is stable, and economy returns to equilibrium over time

New Keynesian dynamics i Y and π both jump down once shock goes away, everything returns to equilibrium no dynamics other than dynamics of shock model is unstable, so economy always in equilibrium if output and inflation do not jump correct amount, off on explosive path

4.1.3 Identification Taylor Rule i t = δ π π t + δ x x t + v t When v t increases, both π t and x t must jump to rule out explosions Therefore error and rhs variables are correlated There are no good instruments as need i t to respond to the variables caused by the error δ π and δ x do not show up in the dynamics of the model

They appear in the roots They need to be large enough for both roots to be unstable (requiring the jump)

5 Monetary Transmission Mechanism Write IS curve as a function of real interest rate gap x t = E t x t+1 1 σ (î t E t π t+1 ) + u t x t = E t x t+1 1 σ (ˆr t r t ) where r t = σu t r t is the Wicksellian real interest rate solve forward x t = 1 σ i=0 E t (ˆr t+i r t+i )

output depends negatively on expected future real interest rate deviations if interest rates are always expected to equal the Wicksellian real rate, then output gaps are zero Where is money in the model? ˆm t ˆp t = ( ) 1 bi ss (σŷ t î t ) money adjusts to get the desired interest rate

6 Economic Disturbances Demand Taste Amend utility function to have a taste shock E t β i ψ1 σ t+i C1 σ t+i + i=0 1 σ Euler equation becomes γ 1 b ( Mt+i P t+i ) 1 b χ N 1+η t+i 1 + η ψt 1 σ Ct σ = βe t ψ 1 σ (1 + i t ) t+1 C σ t+1 1 + π t+1

Linearized around the zero inflation steady state ĉ t = E t ĉ t+1 1 ( ) σ 1 (Et σ (î ) t E t π t+1 ) + ψ t+1 ψ t σ

Government spending Resource constraint changes ŷ t = ( C Y ) ss ĉ t + ( ) G ss ĝ t Y IS curve becomes ˆx t = E tˆx t+1 1 σ (î t E t π t+1 ) + ξ t ξ t = ( ) ( ) σ 1 C ss ( ) ( ) G ss Et ψ t+1 ψ t E t (ĝ t+1 ĝ t ) σ Y Y + ( E t ŷ f t+1 ) ŷf t expected changes matter 1 σ = 1 σ ( ) C ss Y

Supply If no shocks to supply, then stabilization of inflation stabilizes output gap Solving New Keyneisan Phillips Curve forward yields π t = κ t=0 β i E t x t+i Implies that if expected future output gaps are zero, then inflation is at its target of zero Supply shocks change this, raising inflation for each level of output and producing policy tradeoffs What might supply shocks be?

Equation determining inflation before linearization, contained tastes, marginal costs, and productivity Shocks to these variables can add shocks to the Phillips curve However, these shocks constitute shocks to the output gap evaluated at changing values of full-employment output Remember the output gap is evaluated relative to a fixed steady state, so the equilibrium output gap responds to supply shocks Monetary authority should be stabilizing output around the fluctuating output gap, yielding no policy tradeoffs

Sticky wages and prices If only prices are sticky, optimal policy adjusts to keep prices from ever having to adjust (zero inflation) If wages are sticky, and there are real shocks, then a policy which keeps prices from ever having to adjust could increase the output gap Negative productivity shock reduces the marginal product of labor Rigid nominal wages and a monetary policy to keep prices from adjusting would imply MPL(N 0 ) < W P. Firms would reduce employment and output. Now, there is a tradeoff between stabilizing inflation and the output gap.