Foundations for the New Keynesian Model. Lawrence J. Christiano

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 to 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: Constraint on allocation of labor: di, Y i,t e a t N i,t, a t a t t a, t a ~iidn0, a 2 0 Nit di N t

Efficient Allocation of Total Labor Suppose total labor, N t, is fixed. What is the best way to allocate various activities, 0 i? N t among the 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: 0 Nit di 2 2N t 2 2 N t N t

Labor Not Allocated Equally, cnt d Y t Yi,t 0 di 0 2 Y i,t e a t 0 2 di 2 N i,t Yi,t di 2 di Ni,t di e a 2 t 2N t di 0 2 Nt 2 e a 2 t N t 2 di 0 2 2 e a t N t e a t N t f 2 2 2 2 di di

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

Economy with Efficient N Allocation Efficiency dictates So, with efficient production: Resource constraint: N it N t all i Y t e a t N t Preferences: C t Y t E 0 t0 logc t exp t N t, t t t, t ~iid,

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 u n,t u c,t e a t 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 t e a 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 Production frontier 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 profits.

Households Solve: maxe 0 t0 Subject to: t logc t exp t N t, C t bonds purchases in t Bt wage rate wt N t profits t (real) interest on bonds rt 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 0 Pi,t Y i,t di 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 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 Y i,t Subject to demand for : Y i,t P i,t Y t Problem: max N it 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: s t household fonc if = u n,t u c,t expa t. w 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 C t e a t N t exp a t t Bond market clearing 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 t E t C t r 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 t a t t t V, V t 2 a 2 2 2 logr t log E t c t c t a t t t 2 V using assumptions about a t and t log at 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 t = 0.5 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 0.04 9.5 log, r t 0.03 0.02 0.0 logr t log 0.75a t log r t = -log + 0.75a t- log, r t 9 8.5 log r t = -log - (0.5 - ) t /( + ) 0.0 0.009 8 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 employment towards their natural levels.

Introducing Price Setting Frictions (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 Profits, net of taxes raised by Government to finance subsidies. Intratemporal first order condition: 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 E C t R t t C t t E t explogr t log t c t explogr t E t t E t c t, c t logc t take log of both sides: 0 log r t E t t E t c 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 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

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, a t a 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. Its only real decision is to adjust price whenever the opportunity arises.

Intermediate Good Firm Present discounted value of i th firm profits: period tj profits sent to household E t i 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 Conditional expectation over price and aggregate uncertainty (e.g., Δa t and τ t ). Each of the firms that can optimize price choose to optimize P t Conditional expectation over aggregate uncertainty alone. 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.

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.

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 t X 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

Intermediate Good Firm Problem Want p t in: E t j0 j X t,j p t X 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.

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 E t j X t,j j0 s tj s t E t t Kt

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.

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

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. Relationship between aggregate employment and aggregate output not simple because of price distortions: Y t e a t N t, in general This part of the analysis is the reason why it made Calvo famous it s not easy.

Going for Prices Aggregate price relationship P t Pi,t di 0 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.

Key insight P i,t di firms that don t reoptimize price in t add over prices, weighted by # of firms posting that price number of firms that had price, P, in t and were not able to reoptimize in t f t,t P d

Applying the Insight By Calvo randomization assumption f t,t total number of firms with price P in 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!

Expression for p t in terms of aggregate inflation Conclude that this relationship holds between prices: P t P t P t. Only two variables here! Divide by : Rearrange: P t p 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 Yt 0 P i,t P t di Y t P t 0 Pi,t di Y t P t P t Calvo insight Where: P t Pi,t di 0 P t P t

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, all i,j

Example of Efficiency Distortion P j,t P P 2 0 j j. p t P t P t 0.5, 0 P 2 P P 2 P 0.99 0.98 0.97 0.96 0.95 0.94 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 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!

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 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.

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

Solving the Ramsey Problem (surprisingly easy in this case) First, substitute out consumption everywhere 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

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

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.

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

Eventually, Optimal (Ramsey) Equilibrium and Efficient Allocations in Real Economy Coincide Convergence of price distortion 0.98 0.96 0.94 0.92 0.75, 0 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 ) Refer to the Ramsey allocations as the natural allocations. Natural consumption, natural rate of interest, etc.