Foundations for the New Keynesian Model Lawrence J. Christiano
Objective Describe a very simple model economy with no monetary frictions. Describe its properties. markets work well Modify the model dlto include price setting frictions. Now markets won t necessarily work so well, unless monetary policy is good.
Model Household preferences: E 0 t0 t logc t exp t N t, t t t, t ~iidn0, 2
Production Final output requires lots of intermediate inputs: Y t Yi,t 0 Production of intermediate inputs: di, Y a t a a ~iidn0, 2 i,t e N i,t, Δa t Δa t t, t a Constraint on allocation of labor: N it di N t 0
Efficient Allocation of Total Labor Suppose total labor,, is fixed. N t What is the best way to allocate N t among the various activities, 0 i? Answer: allocate labor equally across all the activities N it N t, all i
Suppose Labor Not Allocated Equally Example: N it 2N t i 0, 2 2 N t i,,0. 2 Note that this is a particular distribution of labor across activities: N di 2N it t 2 N N 2 2 t t 0
Labor Not Allocated Equally, cnt d Y t Y i,t 0 di 0 2 Y i,t e a t 0 2 di N i,t Yi,t 2 di 2 di Ni,t di e a 2 t 2N t di 0 2 Nt 2 e a t N t 0 2 2 di 2 2 a t e N t 2 2 2 2 di di e a t N t f
f 2 2 2 2 Efficient Resource Allocation Means Equal Labor Across All Sectors 0 0.98 0.96 0.94 f 0.92 0.9 6 0.88 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 α
Economy with Efficient N Allocation Efficiency dictates N it N t all i So, with efficient production: Y a t e t N t Resource constraint: Preferences: C t Y t E 0 t0 N t logc t t exp t, ~iid, t t, t
Efficient Determination of Labor Lagrangian: max E 0 C t,n t t0 t logc t exp t N t uc t,n t, t t e a t N t C t First order conditions: or: u c C t,n t, t t, u n C t,n t, t t e a t 0 marginal cost of labor in consumption units u n,t u c,t a t u n,t u c,t e 0 du dnt du dct dc t dnt marginal product of labor e a t
Efficient Determination of Labor, cont d Solving the fonc s: u n,t u c,t e a t C t exp t N e a t t e a t N t exp t N t e a t N t exp t Note: C t exp a t t Labor responds to preference shock, not to tech shock
Response to a Jump in a consumption Higher a Indifference curve Budget constraint Leisure, -N
Decentralizing the Model Give households budget constraints and place them in markets. Give the production functions to firms and suppose that they seek to maximize i profits.
Solve: Subject to: Households t N t maxe 0 logc t exp t, t0 bonds purchases in t wage rate profits (real) interest on bonds C t B t w t N t t r t B t First order conditions: u n,t u c,t C t exp t N t w t marginal cost of working equals marginal benefit u c,t E t u c,t r t marginal cost of saving equals marginal benefit
Final Good Firms Final good firms buy Y i,t, i 0,, at given prices,, to maximize profits: P i,t Y t P i,t Y i,t di 0 Subject to Y t Yi,t 0 di Fonc s: P i,t Y t Y i,t Y i,t P i,t Y t, Pi,t di 0
Intermediate Good Firms Y i,t For each there is a single producer who is a monopolist in the product market and hires labor, in competitive labor markets. N i,t Marginal cost of production: (real) marginal costs t dcost dwor ker doutput dwor ker subsidy payment to firm expa t wt t Subsidy will be required to ensure markets work efficiently.
Intermediate Good Firms Demand curve P i,t Y i,t P i,t Y t Marginal cost, s t Marginal revenue Y i,t
ith Intermediate Good Firm Problem: max Nit P it Y it s t Y it Subject to demand for : Problem: max N it Y i,t Y i,t P i,t Y t P it P i,t Y t s t P i,t Y t fonc : P it Y t s t P i,t Y t 0 P it s t price is markup over marginal cost Note: all prices are the same, so resources allocated efficiently across intermediate good firms. P i,t P j,t, because Pi,t di 0
Equilibrium Pulling things together: w s t t expa t household fonc if u n,t u c,t expa t. u n,t u c,t expa t If proper subsidy is provided to monopolists, employment is efficient: if, then u n,t u c,t expa t
Equilibrium Allocations With efficient subsidy, u n,t u c,t functional form Ct exp t N t resource constraint expat exp t N t expa t N t exp t t C a t t e N t exp ep a t Bond marketclearing implies: B t 0 always
Interest Rate in Equilibrium Interest rate backed out of household intertemporal Euler equation: u c,t E t u c,t r t C E t r t C t t r t C E t t E t expc t c t C t E t exp a t a t t t exp E Δa t t, V t 2 a ep t a t V 2 2 2 logr t log E t c t c t Δa t t t 2 V
Dynamic Properties of the Model Response to.0 Technology Shock in Period x 0-3 Response to.0 Preference Shock in Period 0.04-0.5 0.035-0.03 -.5 log C t 0.025 0.02 0.05 0.0 (a t - a t- ) = 0.75(a t- - a t-2 ) + epsa t log C t -2-2.5-3 -3.5-4 τ t = 0.5τ t- + epstau t 0.005-4.5 0 2 4 6 8 0 2 4 6 8 20 22 24 t -5 2 4 6 8 0 2 4 6 8 20 22 24 t interest rate x 0-3 interest rate r t log, r 0.04 0.03 0.02 0.0 0.0 0.009 log r t = -logβ + 0.75a t- 9 t t- r t log, r 9.5 8.5 8 log r t = -logβ - (0.5 - )τ t /( + ) 0.008 7.5 2 4 6 8 0 2 4 6 8 20 22 24 t 2 4 6 8 0 2 4 6 8 20 22 24 t
Key Features of Equilibrium Allocations Allocations efficient (as long as monopoly power neutralized) Employment does not respond to technology Improvement in technology raises marginal product of labor and marginal cost of labor by same amount. First best consumption not a function of intertemporal considerations Discount rate irrelevant. Anticipated future values of shocks irrelevant. Natural rate of interest steers consumption and p employment towards their natural levels.
Introducing Price Setting Frictions (Clarida Gali Gertler 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 N W t C t exp t N t W t P t
Household Intertemporal FONC Condition: or E t u c,t u c,t R t t C t E 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: or 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
Final Good Firms Buy at prices and sell for Y i,t, i 0, P i,t Y t P t Take all prices as given (competitive) Profits: Production function: P t Y t P i,t Y i,t di 0 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
Intermediate Good Firms Each ith good produced by a single monopoly producer. Demand curve: Technology: Y i,t Y t P i,t P t Calvo Price setting Friction 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,
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
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.
Intermediate Good Firm Present tdiscounted d 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 totalt cost P tj s tj Y i,tj Each of the firms that can optimize price choose P to optimize P t in selecting price, firm only cares about future states in which it can t reoptimize E t j j tj P ty t i,tj P tj s tj Y i,tj. j0
Intermediate Good Firm Problem Substitute t out the demand dcurve: E t j tj P ty i,tj P tj s tj Y i,tj j0 j0 E t j tj Y tj P tj P t P tj s tj P t. Differentiate with respect to : or E t j0 j tj Y tj P tj P t P tj s tj P t 0, j P t E t tj Y tj P tj s Ptj tj 0. j0 P t
Intermediate Good Firm Problem Objective: E t j u C tj Y tj P P t tj s tj 0. P tj P tj j0 or j P t E t P tj Ptj s tj 0. j0 E t j0 j X t,j p tx t,j s tj 0, p t P t, X P t,j t tj tj t, j tj tj t, j 0., X t,j X t,j, j 0 t
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 j X t,j j0 s tj K t F t But, still need expressions for K t, F t.
K t E t j X t,j j00 s tj s t E t j X t,j t j t s tj s t E t j X t,j s tj t t j0 E t by LIME s t E t E t t j X t,j s tj j0 exactly K t! s t E t t E t j X t,j j0 0 s tj s t E t t Kt
From previous slide: Kt K t s t E t K t. t 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.
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 F t K t t exp t N t C t expa t E t t Kt. F F j t E t X t,j E t F t t j0
To Characterize Equilibrium Have equations characterizing optimization by firms and households. Still need: Expression for all the prices. Prices, P i,t,0 i, will all be different because of the price setting frictions. i Relationship between aggregate employment and aggregate output not simple becauseof price distortions: Y a t t e N t, in general This part of the analysis is the reason why it made Cl Calvo famous it s not easy.
Going for Prices Aggregate price relationship P t P i,t di 0 P di i,t firms that reoptimize price P i,t firms that don t reoptimize price di 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.
Key insight firms that don t reoptimize price in t P i,t di add over prices, weighted by # of firms posting that price number of firms thatt hd had price, P, in t and were not able to reoptimize i in t f t,t P d
Applying the Insight By Calvo randomization assumption total number of firms with price P in t f t,t f t, for all Substituting: P i,t di firms that don t reoptimize price f t,t P d f t P d P t Something hard got very simple!
p t Expression for in terms of aggregate inflation Conclude that this relationship holds between prices: P t P t P t. Only two variables here! Divide by P : P t Rearrange: p t t t p t t
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.
Tak Yun Algebra Y t 0 Yi,t di 0 At N i,t di labor market clearing At N t demand curve Y P i,t Yt di 0 P t Y t P t Pi,t di 0 Calvo insight Y t P t P t Where: P t Pi,t 0 di P t P t
Relationship Between Agg Inputs and Agg Output Rewriting previous equation: Y t P t P t Yt p t e a t N t, efficiency distortion : p t : P i,t P j,t,, all i,j
Example of Efficiency Distortion P j,t P P 0 j 2 P 2 j. p P t t P P t 0.5, 0 P 2 P 0.99 0.98 0.97 0.96 0.95 0.94 093 0.93 - -0.5 0 0.5 logp /P 2
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)
Equilibrium Conditions Law of motion of (Tak Yun) distortion: p t t t (4) p t Household IntertemporalCondition: E C t t C t R t t (5) Aggregate inputs and output: C t p t e a t N t (6) 6 equations, 8 unknowns:, C t, p t,nn t, t, K t, F t, R t System under determined!
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.
Ramsey Optimal Policy 6 equations in 8 unknowns.. Many configurations i of the 8 unknowns that satisfy the 6 equations. Look kfor the best configurations (Ramsey optimal) Value of tax subsidy and of R represent optimal policy Finding the Ramsey optimal setting of the 6 variables ibl involves solving li a simple Lagrangian optimization problem.
Ramsey Problem max E 0,p t,c t,n t,r t, t,f t,k t t0 0 t logc t exp t N t t E R t t Ct C t C t t C t 2t pt t t p t 3t E t t F t F t 4t C t exp t N t a t e 5t F t t Kt E t t K t K t 6t C t p t e a t N t
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 t logn t logp exp t N t t,p t,n t,r t, t,f t,k t 0 t0 e t p E at R t t N t t e a t Nt 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
Solving the Ramsey Problem, cnt d Simplified problem: max E 0 t logn t logp exp t N t t t,p t,n t t0 2t pt t t p t First order conditions with respect to 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.
Solution to Ramsey Problem Eventually, price distortions eliminated, regardless of shocks p t When price distortions t p gone, so is inflation. t p t p t p t t N t exp Efficient ( first best ) allocations in real economy C t p t e a t N t. Consumption corresponds to efficient allocations in real economy, eventually when price distortions gone
Eventually, Optimal (Ramsey) Equilibrium and Efficient i Allocations in Real Economy Coincide id Convergence of price distortion 0.98 0.96 0.94 0.92 0.75, 0 r p-star 0.9 0.88 p t p t 0.86 0.84 0.82 0.8 2 4 6 8 0 2 4 6 8 20
The Ramsey allocations are eventually the best allocations in the economy without price frictions (i.e., first best allocations ) Rf Refer to the Ramsey allocations as the natural allocations. Natural consumption, natural rate of interest, etc.
Preceding provides important foundations for Preceding provides important foundations for the construction of the New Keynesian model.