Lecture 3, November 30: The Basic New Keynesian Model (Galí, Chapter 3)

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1 MakØk3, Fall 2 (blok 2) Business cycles and monetary stabilization policies Henrik Jensen Department of Economics University of Copenhagen Lecture 3, November 3: The Basic New Keynesian Model (Galí, Chapter 3) c 2 Henrik Jensen. This document may be reproduced for educational and research purposes, as long as the copies contain this notice and are retained for personal use or distributed free.

2 Introduction The classical ex-price model(s) covered so far cannot account for realistic short-run dynamics following monetary shocks In case there are any real e ects, the magnitude appears modest No liquidity e ect present (long run = short run) Some sand in the wheels are necessary to explain the e ects of money on output seen in data Obvious candidate is incomplete nominal adjustment. In ex-price models, money neutrality holds both in the short and long run If money neutrality fails in the short run, the impact of monetary policy will clearly be stronger Classical model is therefore amended with nominal rigidities (and explicit determination of prices by rms) Result is Dynamic Stochastic General Equilibrium (DSGE) model of the New-Keynesian type

3 A basic New-Keynesian model Goods market Demand side: Households consume a basket of goods (based on utility maximization) Supply side: Firms produce di erent consumption goods (maximize pro ts under monopolistic competition) Labor market Demand side: Firms hire labor (maximize pro ts under monopolistic competition) Supply side: Households supply labor (based on utility maximization) Financial markets Households optimally invest in a one-period risk-less bond Households also hold money. We again just assume it (and generally do not make much use of it here, as focus will be on interest-rate policies) 2

4 Wages are perfectly exible (this assumption is relaxed in Chapter 6) All goods prices are subject to in exibility, i.e., stickiness: Every period, each rm faces a state-independent probability of being stuck with its price So, is natural index for stickiness in the economy The probability is independent of when the rm last changed its price; i.e., time-dependent price stickiness (as opposed to state-dependent price stickiness) Stylistic representation of staggering. The independence of history facilitates aggregation: is the fraction of rms not adjusting prices in a period is the fraction of rms adjusting is a period :::: = = ( ) is average duration of a price contract When allowed to change the price, expectations about future prices become of importance 3

5 A view at prices changes: Belgium, 2-25 Source: Cornille and Dossche (26) 4

6 A view at prices changes: France and Italy, Source: Dhyne et al. (25) 5

7 A view at prices changes: (Monthly) frequencies of price adjustments; Euro area vs. US Source: Dhyne et al. (25) (with US data based on Bils and Klenow, 24) 6

8 A view at prices changes: France, hazard rates in clothing industry Source: Baudry et al. (24) 7

9 A view at prices changes: France: Hazard Rates in service industry Source: Baudry et al. (24) 8

10 Summary of ndings from data: Aggregate prices are sticky (as also indicated by VAR analyses) Price setting in di erent sectors di ers; some sectors exhibit long periods of xed prices (time dependent price setting) some sectors exhibit continuous changing prices Price setting is asynchronized across sectors In some sectors, the probability of price adjustment is (nearly) independent of the life-span of the price Relation to theoretical modelling of price rigidities: The staggered price setting scheme captures many of these features, and their aggregate implications, in a simple and handy way Modelling due to Calvo (983); now just called Calvo pricing 9

11 Household behavior The essentials of household behavior are like in the basic classical model Household maximize expected utility E X t= t U (C t ; N t ) ; < < : Main di erence: C t is not one good, but a basket of the di erent goods produced by the di erent producers (indexed by i 2 [; ]) Z " C t C t (i) " " " di ; " > Budget constraint therefore becomes: Z P t (i) C t (i) di + Q t B t B t + W t N t T t (and a No-Ponzi Game constraint ruling out explosive debt) Household chooses optimal paths of C t, N t and B t (taking prices P t, W t and Q t as given), but must now choose the composition of C t as well

12 Finding the optimal composition of C t : Assume that total nominal expenditures on consumption are Z t Z P t (i) C t (i) di = Z t Optimal demand for C t (i) ; is found by max C t (i) Z C t (i) " " di " " s.t. Z P t (i) C t (i) di = Z t The relevant rst-order condition is Z C t (i) " " di Z " C t (i) " = P t (i) ; C t (i) " " " di C t (i) " " = P t (i) C t (i) where is the Lagrange multiplier associated with the constraint

13 Integrating over all goods: Z Z C t (i) " " " Z di C t (i) " " di = P t (i) C t (i) di C t = Z t Let P t be the price index associated with C t, then P t C t = Z t and Inserted into rst-order condition: Z The relative demand for good i is therefore = P t C t (i) " " " di C t (i) " = P t (i) ; P t C " t C t (i) " = P t (i) P t C t (i) = " Pt (i) C t () P t Note that " is the elasticity of substitution between goods. competition applies when "!.) (Hence, the limiting case of perfect 2

14 The price index The precise form of the price index P t follows from and the expression for relative demand P t C t = Z P t (i) C t (i) di I.e., we get P t C t = P t = Z " Pt (i) P t (i) C t di Z P t (i) P t " di " 3

15 Remaining household optimality conditions Precisely as in the simple classical model: U n;t U c;t = W t P t Q t = E t Uc;t+ U c;t P t P t+ With U (C t ; N t ) C t Labor supply schedule: N +' t +', ; ' >, we have the same log-linear expressions: c t + 'n t = w t p t (2) Consumption-Euler equation: with i t log Q t and log c t = E t fc t+ g (i t E t f t+ g ) (3) 4

16 Firms behavior and aggregate in ation dynamics Generally, optimal price-setting of a rm that can change its price in period t: X max k E t Qt;t+k P Pt t Y t+kjt t+k Y t+kjt k= P " s.t. Y t+kjt = t C t+k (8) P t+k I.e., the maximal expected discounted stream of pro ts of choosing Pt probabilities that it is xed in the future t+k is total costs of sales at price P t taking into account the in period t + k (will depend on labor costs and productivity) Q t;t+k is the stochastic discount factor (or present-value operator), equal to the one facing consumers between periods t and t + k: Q t;t+k k Ct+k Pt C t P t+k Note Q t;t+ Q t;t+2 Ct+ Pt = C t P t+ = 2 Ct+2 Pt = C t P t+2 Ct+ C t Pt Q t;t+k = Q t;t+ Q t+;t+2 :::::Q t+k 2;t+k Q t+k ;t+k P t+ Ct+2 C t+ Pt+ = Q t;t+ Q t+;t+2 P t+2 5

17 First-order condition for Pt : X k E t Q t;t+k k= Y t+kjt + P t where t+kjt t+k Y t+kjt is nominal marginal costs X k E t Q t;t+k Y t+kjt + P t+kjt Y k= t+kjt Y X k " E t Q t;t+k Y t+kjt " + t+kjt Pt k= We then get the central pricing equation: X k E t Qt;t+k Y t+kjt with M "= (" k= = ; = ; = : P t M t+kjt = ; (9) ) > being the mark-up; a measure of the monopoly distortion 6

18 This is rewritten as X k= P k E t Q t;t+k Y t t+kjt MMC t+kjt t ;t+k P t = ; () where MC t+kjt t+kjt =P t+k is real marginal costs and t ;t+k P t+k =P t is gross in ation between t and t + k Optimality condition is log-linearized around a zero-in ation steady state: Pt = ; t ;t+k = ; Y t+kjt = Y; MC = M ; Q t;t+k = k P t One gets X () k E t p t p t cmc t+kjt p t+k + p t = k= where cmc t+kjt mc t+kjt mc = mc t+kjt + log M = mc t+kjt + log " " = mc t+kjt + Hence, X () k p t = k= X () k E t cmct+kjt + p t+k k= p t = ( ) X () k E t cmct+kjt + p t+k k= Optimal price is a function of expected current and future real marginal costs and aggregate prices 7

19 Marginal costs depend on production function and factor payments Each rm has the following production function (i.e., identical technology): Y t (i) = A t N t (i) ; < (5) Total costs T C t = W t N t (i) Real marginal costs MC t (i) = P t A t ( W t ) N t W t P t MP N t = W t P t A t ( ) (Y t (i) =A t ) = W t P t ( ) A t Y t (i) A Special case of constant returns to scale, = MC t+kjt = MC t+k = W t+k P t+k A t+k X p t = ( ) () k E t f cmc t+k + p t+k g k= This is the unique stationary solution to the rst-order rational expectations di erence equation: p t = E t fp t+g + ( ) ( cmc t + p t ) 8

20 Aggregate price dynamics Remember, is fraction of price setters that keep past period s price; is fraction of price setters that set new prices By de nition of the price index: h P t = (P t ) " + ( ) (Pt ) "i " Log-linearized around the zero-in ation steady state: P =) t " " = + ( ) t P t ( ") t = ( ") ( ) (p t p t ) =) t = ( ) (p t p t ) (7) Rewrite the di erence equation for optimal price setting: Use aggregate dynamics to obtain: p t p t = E t fp t+ p t g + ( ) ( cmc t + p t ) p t + p t p t p t = E t fp t+ p t g + ( ) cmc t + t ( ) t = ( ) E t f t+ g + ( ) cmc t + t ( ) ( ) t = E t f t+ g + cmc t ; ; lim = :! This is the basic in ation-adjustment equation in New-Keynesian theory 9

21 General case of decreasing returns to scale, < < We de ne aggregate real marginal costs (in logs): We have for a price-changing rm: mc t w t p t mpn t = w t p t (a t y t ) log ( ) mc t+kjt = w t+k p t+k a t+k y t+kjt = mc t+k + y t+kjt y t+k log ( ) By the demand function in logs (here using equilibrium condition y t+k = c t+k ): mc t+kjt = mc t+k " (p t p t+k ) (4) Optimal price setting: p t = ( Associated in ation dynamics ) X k= t = E t f t+ g + cmc t ; () k " E t cmc t+k (p t p t+k ) + p t+k ( ) ( ) + " ; lim! = : (6) 2

22 Equilibrium outcomes Goods market clearing on each goods market Y t (i) = C t (i) ; all i In the aggregate: where Y t Z Y t = C t ; " Y t (i) " " " di : Asset market equilibrium (as in classical economy): y t = E t fy t+ g (i t E t f t+ g ) (2) 2

23 Aggregate employment: N t = Z N t (i) di = Z Yt (i) A t di N t = Yt A t Yt Z Z Yt (i) Y t Pt (i) di " di In logs: A t P t n t = Z (y t a t ) + log Pt (i) ( ) n t = y t a t + d t ; d t ( ) log P t " ; Z Pt (i) P t " : The variable d t is a natural measure of price dispersion ( proportional to the variance of prices) When we consider log deviations from a symmetric zero-in ation steady state, d t can be ignored (it is of second order see Appendix 3.2 and 3.3) Hence, as a log-linear approximation: as in the classical case y t = a t + ( ) n t ; (3) 22

24 Relationship between real marginal cost and output We want a dynamic system in output and in ation; hence, marginal cost is replaced appropriately by output We have mc t = w t p t (a t y t ) log ( ) Using the labor supply schedule mc t = c t + 'n t (a t y t ) log ( ) mc t = y t + ' (y t a t ) ( ) + ' + = y t Under exible prices, =, cmc t = mc t = mc (a t y t ) log ( ) + ' a t log ( ) (7) = (= ln M) Hence, denoting y n t the exible-price output, or, the natural rate of output, mc = = ( ) + ' + yt n + ' a t log ( ) 23

25 The natural rate of output is given as yt n + ' = ( ) + ' + a [ log ( )] ( ) t ( ) + ' + = n yaa t + # n y (9) Under perfect competition, =, this is output as in the classical model We get the marginal cost in deviation from steady state: cmc t = mc t mc = We denote ey t y t y n t the output gap ( ) + ' + (y t y n t ) (2) The New-Keynesian Phillips Curve ( NKPC ): ( ) + ' + t = E t f t+ g + ey t ; (2) 24

26 Rephrase the Euler equation in terms of the output gap: y t = E t fy t+ g (i t E t f t+ g ) ey t = E t fey t+ g (i t E t f t+ g ) yt n + E t fyt+g n ey t = E t fey t+ g (i t E t f t+ g rt n ) (22) where is the natural rate of interest r n t + E t fy n t+g A dynamic IS curve ( DIS ) Let br n t r n t New-Keynesian Phillips curve and dynamic IS curve determines output gap and in ation conditional on monetary policy (i t ). So, in contrast with the classical model, monetary policy matters for output determination! 25

27 Interest rate policy in the New Keynesian model The case of a simple Taylor-type interest rate rule: i t = + t + y ey t + t ; ; y ; (25) where t is a policy shock, v t = v v t + " v t The policy rule, NKPC and DIS give the dynamics eyt Et fey = A t+ g T + B t E t f t+ g T (br t v t ) ; (26) A T ; B + + T ; y + + y A unique non-explosive equilibrium requires the eigenvalues of A T to be stable (within the unit circle) Uniqueness depends on policy-rule parameters: < ( ) + y ( ) (27) is necessary and su cient condition (the Taylor principle ) Note > is su cient condition (an active Taylor rule) 26

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30 Concluding remarks The New-Keynesian model is (compared to the classical model) a simple but powerful tool for analyzing monetary policy Given monetary policy works more in accordance with evidence, how can the model be used for policy evaluation? To come: Welfare e ects of di erent policies Optimal monetary policy and credibility issues 29

31 Next time(s) Monday, December 6: Exercises: Derive the dynamics of the New-Keynesian model on matrix form (or, state-space form), (26) Derive the condition for determinacy, (27) in Galí, Chapter 3. Hint: A 2 2 matrix has two stable roots, when the coe cients in the characteristic polynomial 2 + a + a satisfy ja j < and ja j < + a Derive the explicit solutions for the output gap, in ation, and the nominal interest rate and output in New-Keynesian model of Galí, Chapter 3 (page 5) under an interest-rate rule (and in the absence of technology shocks). Interpret the solutions economically Tuesday, December 7: Lecture: Monetary policy design and welfare in the simple NK Model (Galí, Chapter 4) 3