Simple New Keynesian Model without Capital. Lawrence J. Christiano
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1 Simple New Keynesian Model without Capital Lawrence J. Christiano
2 Outline Formulate the nonlinear equilibrium conditions of the model. Need actual nonlinear conditions to study Ramsey optimal policy, even if we want to use linearization methods to study Ramsey. Ramsey will be used to define output gap in positive model of the economy, in which monetary policy is governed by the Taylor rule. Later, when discussing timeless perspective, will discuss use of Ramsey optimal policy in actual, real time implementation of monetary policy. Need nonlinear equations if we were to study higher order perturbation solutions. Study properties of the NK model with Taylor rule, using Dynare.
3 Clarida Gali Gertler Model Households maximize: E 0 t0 Subject to: logc t exp t N t, t t t, t ~iid, P t C t B t W t N t R t B t T t Intratemporal first order condition: C t exp t N t W t P t
4 Household Intertemporal FONC Condition: E t u c,t u c,t R t t or, for when we do linearize later: E C t R t t C t t E t explogr t log t Δc t explogr t E t t E t Δc t, c t logc t take log of both sides: 0 log r t E t t E t Δc t, r t logr t or c t log r t E t t c t
5 Final Good Firms Buy at prices and sell for Take all prices as given (competitive) Profits: Y i,t, i 0, P i,t Y t P t Production function: P t Y t 0 Pi,t Y i,t di Y t Yi,t 0 di,, First order condition: Y i,t Y t P i,t P t P t Pi,t di 0
6 Intermediate Good Firms Each ith good produced by a single monopoly producer. Demand curve: Technology: Y i,t Y t Calvo Price setting Friction P i,t P t Y i,t expa t N i,t, Δa t Δa t t a, P i,t P t P i,t with probability with probability,
7 Marginal Cost real marginal cost s t dcost dwor ker doutput dwor ker W t/p t expa t in efficient setting C t exp t N t expa t
8 The Intermediate Firm s Decisions ith firm is required to satisfy whatever demand shows up at its posted price. It s only real decision is to adjust price whenever the opportunity arises.
9 Intermediate Good Firm Present discounted value of firm profits: period tj profits sent to household E t j0 j marginal value of dividends to householdu c,tj /P tj tj revenues P i,tj Y i,tj total cost P tj s tj Y i,tj Each of the firms that can optimize price choose to optimize P t E t j0 j in selecting price, firm only cares about future states in which it can t reoptimize j tj P ty i,tj P tj s tj Y i,tj.
10 Intermediate Good Firm Problem Substitute out the demand curve: Differentiate with respect to : or E t j tj P ty i,tj P tj s tj Y i,tj j0 E t j tj Y tj P tj P t P tj s tj P t. j0 P t E t j tj Y tj P tj P t P tj s tj P t 0, j0 E t j tj Y tj P tj j0 P t P tj s tj 0.
11 Intermediate Good Firm Problem Objective: or E t j u C tj P tj j0 E t j P tj j0 Y tj P tj P t P tj s tj 0. P t P tj s tj 0. E t j X t,j p tx t,j s tj 0, j0 p t P t P t, X t,j tj tj t, j, j 0., X t,j X t,j t, j 0
12 Intermediate Good Firm Problem Want p t in: E t j0 j X t,j p tx t,j s tj 0 Solution: p t E t j0 j X t,j E t j0 j X t,j s tj K t F t But, still need expressions for K t, F t.
13 K t E t j X t,j j0 s tj s t E t j X t,j t j s t E t t s t E t by LIME E t E t t j X t,j j0 s tj s tj j X t,j j0 exactly K t! s tj s t E t t s t E t t Kt E t j X t,j j0 s tj
14 From previous slide: K t s t E t t Kt. Substituting out for marginal cost: dcost/dlabor s t W t /P t doutput/dlabor expa t W t Pt by household optimization exp t N t C t expa t.
15 In Sum solution: Where: p t E t j0 j X t,j E t j0 j X t,j s tj K t F t, K t t exp t N t C t expa t E t t Kt. F t E t j X t,j E t t j0 Ft
16 To Characterize Equilibrium Have equations characterizing optimization by firms and households. Still need: P i,t,0 i Expression for all the prices. Prices,, will all be different because of the price setting frictions. Relationship between aggregate employment and aggregate output not simple because of price distortions: Y t e a t N t, in general
17 Going for Prices Aggregate price relationship P t Pi,t di 0 Calvo insight: This is just a simple function of last period s aggregate price because non-optimizers chosen at random. P i,t di firms that reoptimize price P i,t di firms that don t reoptimize price all reoptimizers choose same price P t P i,t di firms that don t reoptimize price In principle, to solve the model need all the prices, P t, P i,t,0 i Fortunately, that won t be necessary.
18 Expression for in terms of aggregate inflation Conclude that this relationship holds between prices: P t p t P t P t. Only two variables here! Divide by : P t Rearrange: p t t p t t
19 Relation Between Aggregate Output and Aggregate Inputs Technically, there is no aggregate production function in this model If you know how many people are working, N, and the state of technology, a, you don t have enough information to know what Y is. Price frictions imply that resources will not be efficiently allocated among different inputs. Implies Y low for given a and N. How low? Tak Yun (JME) gave a simple answer.
20 Tak Yun Algebra Y t 0 Yi,t di 0 At N i,t di labor market clearing At N t demand curve Yt 0 P i,t P t di Y t P t 0 Pi,t di Calvo insight Y t P t P t Where: P t Pi,t di 0 P t P t
21 Relationship Between Agg Inputs and Agg Output Rewriting previous equation: Y t P t P t Yt efficiency distortion : p t e a t N t, p t : P i,t P j,t,alli,j
22 Collecting Equilibrium Conditions Price setting: K t exp t N t C t A t E t t K t () F t E t t F t (2) Intermediate good firm optimality and restriction across prices: p t by firm optimality Kt F t p t by restriction across prices t (3)
23 Equilibrium Conditions Law of motion of (Tak Yun) distortion: p t t t p t (4) Household Intertemporal Condition: E C t t C t Aggregate inputs and output: 6 equations, 8 unknowns: R t t (5) C t p t e a t N t (6), C t, p t,n t, t, K t, F t, R t System under determined!
24 Underdetermined System Not surprising: we added a variable, the nominal rate of interest. Also, we re counting subsidy as among the unknowns. Have two extra policy variables. One way to pin them down: compute optimal policy.
25 Ramsey Optimal Policy 6 equations in 8 unknowns.. Many configurations of the 8 unknowns that satisfy the 6 equations. Look for the best configurations (Ramsey optimal) Value of tax subsidy and of R represent optimal policy Finding the Ramsey optimal setting of the 6 variables involves solving a simple Lagrangian optimization problem.
26 Ramsey Problem max E 0,p t,c t,n t,r t, t,f t,k t t0 t logc t exp t N t t Ct E t C t 2t pt R t t t t p t 3t E t t F t F t 4t 5t F t t C t exp t N t e a t Kt E t t K t K t 6t C t p t e a t N t
27 Solving the Ramsey Problem (surprisingly easy in this case) First, substitute out consumption everywhere defines R defines F defines tax defines K max E 0,p t,n t,r t, t,f t,k t t0 e t p E at R t t N t t e a t Nt t logn t logp t exp t N t p t 2t pt t t t p t 3t E t t F t F t 4t exp t N t p t E t t K t K t 5t F t t Kt
28 Solving the Ramsey Problem, cnt d Simplified problem: max E 0 t,p t,n t t0 t logn t logp t exp t N t 2t pt t First order conditions with respect to t p t p t, t, N t p t 2,t t 2t, t p t p t, Nt exp t Substituting the solution for inflation into law of motion for price distortion: p t p t.
29 Solution to Ramsey Problem Eventually, price distortions eliminated, regardless of shocks When price distortions gone, so is inflation. Efficient ( first best ) allocations in real economy p t t p t p t p t N t exp t C t p t e a t N t. Consumption corresponds to efficient allocations in real economy, eventually when price distortions gone
30 Eventually, Optimal (Ramsey) Equilibrium and Efficient Allocations in Real Economy Coincide Convergence of price distortion , 0 p-star p t p t
31 The Ramsey allocations are eventually the best allocations in the economy without price frictions (i.e., first best allocations ) Refer to the Ramsey allocations as the natural allocations. Natural consumption, natural rate of interest, etc.
32 Equations of the NK Model Under the Optimal Policy ( Natural Equilibrium ) Output and employment is (eventually) y t a t t, n t t Intertemporal Euler equation after taking logs and ignoring variance adjustment term: y t r t E t t rr E t y t, rr log Inflation in Ramsey equilibrium is (eventually) zero.
33 Solving for Natural Rate of Interest Intertemporal euler equation in natural equilibrium: y t y t a t t r t rr E t Back out the natural rate: a t t Shocks: r t rr Δa t t t t t, Δa t Δa t t
34 Next, Put Turn to the NK Model with Taylor Rule
35 Taylor Rule Taylor rule: designed, so that in steady state, inflation is zero ( ) Employment subsidy extinguishes monopoly power in steady state:
36 NK IS Curve Euler equation in two equilibria: Taylor rule equilibrium: y t r t E t t rr E t y t Natural equilibrium: y t r t rr E t y t Subtract: Output gap x t r t E t t r t E t x t
37 Output in NK Equilibrium Agg output relation: y t logp t n t a t,logp t 0 ifp i,t P j,t for all i, j 0 otherwise. To first order approximation, p t p t 0 t, p t
38 Price Setting Equations Log linearly expand the price setting equations about steady state. E t t F t F t 0 F t t K t 0 C t exp t N t e a t Log linearly expand about steady state: t E t t K t K t 0 x t t See
39 Taylor Rule Policy rule r t r t rr t x x t u t,,x t y t y t
40 Equations of Actual Equilibrium Closed by Adding Policy Rule E t t x t t 0 (Phillips curve) r t E t t r t E t x t x t 0 (IS equation) r t t x x t r t 0(policyrule) r t Δa t t 0 (definition of natural rate)
41 Solving the Model s t Δa t t 0 0 Δa t t t t s t Ps t t t 0 0 t x t r t 0 x 0 x t r t r t rr t t x t r t s t s t r t E t 0 z t z t 2 z t 0 s t s t 0
42 Solving the Model E t 0 z t z t 2 z t 0 s t s t 0 Solution: As before: z t Az t Bs t 0 A 2 A 2 I 0, s t Ps t t 0. F 0 0 BP 0 A B 0
43 x 0,. 5, 0.99,, 0.2, 0.75, 0, 0. 2, 0.5. Dynamic Response to a Technology Shock inflation output gap 0.2 nominal rate natural nominal rate actual nominal rate natural real rate employment log, technology natural employment actual employment output natural output actual output
44 Dynamic Response to a Preference Shock inflation output gap nominal rate natural real rate actual nominal rate natural real rate preference shock output natural output actual output employment natural employment actual employment
45 Conclusion of NK Model Analysis We studied examples in which the Taylor rule moves the interest rate in the right direction in response to shocks. However, the move is not strong enough. Will consider modifications of the Taylor rule using Dynare.
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