Lecture 6, January 7 and 15: Sticky Wages and Prices (Galí, Chapter 6)

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1 MakØk3, Fall 2012/2013 (Blok 2) Business cycles and monetary stabilization policies Henrik Jensen Department of Economics University of Copenhagen Lecture 6, January 7 and 15: Sticky Wages and Prices (Galí, Chapter 6) c 2013 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 Introductory remarks In the simple New-Keynesian model nominal rigidities takes the form of goods price rigidities A crucial break with the classical model of (Galí, 2008, Chapter 2), as monetary policy then plays a role for output determination Evidence pointed to the realism of presence of goods price rigidities The model s labor market is, however, portrayed as classical in the sense that a exible nominal wage clears the labor market Unrealistic in itself Precludes, by construction, meaningful talk about unemployment in absence of other labor market frictions The model of Chapter 6, however, does not explicitly consider unemployment, but Galí shows how it can be restated to account for unemployment in his 2011 book 1

3 Source: Knell (2010, ECB WP) 2

4 The basic model is therefore extended with nominal wage rigidity To understand the implications of nominal wage rigidity, it is introduced in a way akin to goods price rigidities This re ects how a macroeconomic research program typically develops Start with a simple model If it creates only nonsense, stop the program. If it gives some insights the profession nds valuable in terms of empirical validity and/or policy prescriptions, then continue Extend the simple model with further realistic assumptions (i.e., relax some of the simplifying assumptions) Indeed, the introduction of sticky wages came in 2000 (by Erceg, Henderson and Levin), after the basic model was developed in mid-1990s (by King and Wolman, 1996; Yun, 1996; Woodford, 1996; Rotemberg and Woodford, 1997). Made into medium-scale models by, e.g., Smets and Wouters (2003) and Christiano et al. (2005, JPE) Ex post, this gradual evolution has pedagogical advantages one learns better how each extension contributes to the model s properties A medium-scale estimated New-Keynesian model as RAMSES used for monetary policy analysis in Sweden, would be impossible to explain from scratch 3

5 The basic NK model with sticky wages Most assumptions from the basic model of Chapter 3 are retained New assumptions: Regarding the labor market: Each household supplies a distinct type of labor Each type of labor is used by the rms Each household has monopoly power in the determination of their nominal wage (or, are represented by a trade union) labour suppliers are therefore monopolistic competitors; just as good suppliers ( rms) Nominal wage setting is subject to rigidities: Independent of past wage history, a wage setter cannot reset its wage with probability w 2 [0; 1] Regarding the asset market: Complete markets are assumed (see my expository note on web) Existence of a complete set of state-contingent securities implies full income insurance, so marginal utility of income is equalized across households 4

6 Optimal behavior by rms For each rm i 2 [0; 1] (producing a unique consumption good), production function is where N t (i) Z 1 0 Y t (i) = A t N t (i) 1 (1) N t (i; j) " w 1 "w "w 1 "w dj with N t (i; j) being rm i s employment of type-j labor, j 2 [0; 1]. ; " w > 1 (2) In addition to choosing a price (if allowed by the Calvo mechanism) and production, a rm must choose the optimal composition of the labor it employs Let W t (j) be the nominal wage of type-j labor Optimization is in two stages: 1) For any choice of N t (i) choose the N t (i; j)s taking as given the nominal wages 2) Choose optimal prices as in Chapter 3 (equivalent of choosing N t (i)) Remark: This mirrors households consumption choice: First, they choose the composition of consumption goods, then it chooses the consumption aggregate 5

7 The optimal choice of N t (i; j)s follows by cost minimization as "w Wt (j) N t (i; j) = N t (i) ; all i; j 2 [0; 1] (3) where W t W t Z 1 W t (i) 1 0 " w 1 1 "w Remark the analogy, or symmetry, with the optimal consumption decisions of households derived in Chapter 3 (Appendix 3.1): "p Pt (i) C t (i) = C t (Eq.1, Chap. 3) where and where " is replaced by P t = P t Z 1 P t (i) " 1 "p p di * " p > 1, the elasticity of substitution between types of goods, distinct from * " w > 1, the elasticity of substitution between types of labor (4) We also get Z 1 W t (j) N t (i; j) dj = W t N t (i), analogy to : 0 Z 1 0 P t (i) C t (i) di = P t C t 6

8 Second step of rm s optimization: Reset price according to max k pe t Qt;t+k P Pt t Y t+kjt t+k Y t+kjt where has been replaced by s.t. Y t+kjt = P "p t C t+k (Eq. 8, Chap. 3) P t+k * p, the probability of rms not being able to reset price, distinct from * w, the probability households not being able to reset wage In Chapter 3 we saw that log-linearized around a zero-price-in ation steady state this gave the optimal (log) price as p t = p + (1 p ) ( p ) k E t mct+kjt + p t+k where p log M p = log " p " p 1 = mc is log of the desired price markup This was shown to give the following in ation-adjustment equation: p t = E t p t+1 + p cmc t ; p (1 p) (1 p ) 1 (Eq. 16, Chap. 3) p 1 + " p with p t replacing t to emphasize that it is price in ation 7

9 To facilitate comparison with upcoming equation for wage dynamics, price dynamics are written in terms of markup uctuations instead of marginal cost uctuations: p t = E t p t+1 + p cmc t = E t p t+1 + p (mc t mc) = E t p t+1 + p (mc t + p ) E t p t+1 p ( p t p ) = E t p t+1 p b p t (5) Optimal behavior by households Households, now indexed by j 2 [0; 1], has utility ( 1 ) X E 0 t U (C t (j) ; N t (j)) t=0 with the usual consumption index (already used for relative demand schedules in rms optimization): Z 1 " p C t C t (i) " p 1 "p 1 "p di 0 The new aspect is that each household supplies the unique labor type j, and sets a nominal wage taking into account the rms relative demand for labor types and the wage-calvo assumption 8

10 A household that resets its wage in period t will choose Wt to maximize ( X 1 E t ( w ) k ) U C t+kjt ; N t+kjt subject to and budget constraint: N t+kjt = W "w Z 1 t N t+k; N t+k N t+k (i) di W t+k 0 P t+k C t+kjt + E t+k Qt+k;t+k+1 D t+k+1jt D t+kjt + W t N t+kjt T t+k where D t+kjt is market value of portfolio of state-contingent claims, and Q t+k;t+k+1 is the stochastic discount factor (see expository note for details) 9

11 First-order condition for Wt : ( w ) k E t U c ( w ) k E t U c C t+kjt ; N t+kjt 1 P t+kjt C t+kjt ; N N t+kjt + W ( w ) k E t N t+kjt U c C t+kjt ; N t+kjt 1 P t+k ( w ) k E t N t+kjt U c ( w ) k E t N t+kjt U c where M w " w " w 1 Rewritten as ( w ) k E t N t+kjt U c Wt C t+kjt ; N t+kjt P t+k ( w ) k E t N t+kjt U t+kjt + U n C t+kjt ; N 1 t N t+kjt 1 t+kjtw t N t+kjt = t+kjt + U n C t+kjt ; N = t+kjt 1 + U n C t+kjt ; t+kjt + U n C t+kjt ; N W t C t+kjt ; N t+kjt (1 " w ) " w U n C t+kjt ; N t+kjt P t+k > 1 is the desired wage markup Wt C t+kjt ; N t+kjt P t+k Wt C t+kjt ; N t+kjt + M w U n P t+k 10 M w MRS t+kjt C t+kjt ; N t+kjt = 0 = 0 N t+kjt W t N t+kjt = 0 = 0 = 0; MRS t+kjt U n C t+kjt ; N t+kjt U c C t+kjt ; N t+kjt : (8)

12 In special case of full wage exibility: W t P t = W t P t = M w MRS t ; i.e., wages are set as a markup over the marginal rate of substitution (introducing an ine ciently high real wage due to the monopoly power of households) First-order condition is log-linearized around a zero-wage-in ation steady state, and one gets the (log) optimal wage: wt = w + (1 w ) ( w ) k E t mrst+kjt + p t+k (9) where w log M w : Remark the analogy with the previously stated optimal price-setting rule p t = p + (1 p ) ( p ) k E t mct+kjt + p t+k We can therefore derive an analogous wage-in ation schedule 11

13 With the assumed utility function, U (C; N) = 1 1 C ' N 1+' ; MRS = U n U c = C N ' : Hence, mrs t+kjt = c t+kjt + 'n t+kjt Due to the assumption about complete asset markets, c t+kjt = c t+k Hence mrs t+kjt = c t+k +'n t+kjt, and when the average marginal rate of substitution in the economy is de ned as mrs t+k = c t+k + 'n t+k one gets mrs t+kjt = mrs t+k + ' n t+kjt n t+k = mrs t+k " w ' (w t w t+k ) Note the analogy with the marginal cost expression used for the derivation of price dynamics mc t+kjt = mc t+k " p 1 (p t p t+k ) (Eq. 14, Chap. 3) 12

14 The optimal (log) wage then becomes wt = w + (1 w ) ( w ) k E t fmrs t+k " p ' (wt w t+k ) + p t+k g w t 1 + (1 w )! ( w ) k " p ' = w + (1 w ) w t (1 + " p ') = w + (1 w ) wt = 1 w 1 + " p ' ( w ) k E t fmrs t+k + " p 'w t+k + p t+k g ( w ) k E t fmrs t+k + " p 'w t+k + p t+k g ( w ) k E t f w + mrs t+k + " p 'w t+k + p t+k g Let w t w t p t mrs t be the economy s average wage markup. Then, wt = 1 w ( w ) k E t f w + w t+k p t+k w t+k + " p 'w t+k + p t+k g 1 + " p ' = 1 w ( w ) k E t f(1 + " p ') w t+k b w 1 + " p ' t+kg where b w t w t w. This is the unique stationary solution to the rst-order rational expectations di erence equation: wt = w E t fwt+1g + (1 w ) w t (1 + " p ') 1 b w t (10) 13

15 Combined with the log-linear dynamics for aggregate wages from wage index, w t = w w t 1 + (1 w ) w t ; (11) one gets a wage in ation equation w t = E t f w t+1g w b w t ; w (1 w) (1 w ) w (1 + " w ') where w t w t w t 1 is nominal wage in ation (12) Note the analogy with the price in ation curve Note that shocks to the economy under wage rigidity cause uctuations in the wage markup (the wedge between real wages and the marginal rate of substitution), and thus variations in wage in ation [so (12) replaces the condition w t p t = mrs t in the model with exible wages] Optimal intertemporal allocation of consumption across time is independent of wage setting and is given by c t = E t fc t+1 g 1 i t E t p t+1 (13) 14

16 Equilibrium Again, the equilibrium will be formulated in terms of gaps The output gap, ey t y t yt n is again output relative to the natural rate; but now yt n is output under exible prices and exible nominal wages The real wage gap is de ned as with the real wage being e! t =! t! n t! t w t p t The natural real wage is found from the de nition of marginal costs: which de nes! n t as mc t w t p t mpn t =! t (y t n t ) log (1 )! n t = log (1 ) + (y n t n n t ) p = log (1 ) + n yaa t 1 1 n yaa t a t = log (1 ) + n waa t p ; n wa 1 n ya 1 p 15

17 We now express the price Phillips curve in terms of gaps We again use mc t w t p t mpn t but expressed as average markup (mc t = p t): p t = mpn t! t Since! n t = log (1 ) + (y n t n n t ) p we get b p t = mpn t! t p b p t = y t n t! t p + log (1 ) b p t = (ey t en t ) e! t = 1 ey t e! t (14) Inserted into price in ation equation (5): p t = E t p t+1 + p ey t + p e! t ; p p 1 (15) 16

18 Similarly, to express the wage Phillips curve in terms of gaps, we use that and thus w t = w t p t mrs t b w t =! t mrs t w = e! t (ey t + 'en t ) = e! t + ' ey t : (16) 1 Inserted into the wage-in ation curve (12): w t = E t f w t+1g + w ey t + w e! t ; w w + ' 1 (17) 17

19 In addition to the two Phillips curves we have the usual Euler-equation written in terms of the output gap: ey t = E t fey t+1 g 1 i t E t p t+1 rt n (19) where is the natural rate of interest r n t + E t fy n t+1g The model is closed by a speci cation of monetary policy in the form of a generalized Taylor rule: i t = + p p t + w w t + y ey t + v t and a de nition equation linking real wages and price and nominal wage in ation: e! t = e! t 1 + w t p t! n t We now have ve equations to solve for the ve endogenous variables ey t, p t, w t, e! t, i t as functions of the exogenous shocks (a t and v t ), and given e! t 1 18

20 To simplify slightly, the nominal interest rate rule is inserted into the Euler-equation, and the system of four equilibrium conditions are written in matrix form as A w;0 x t = A w;1 E t fx t+1 g + B w z t x t [ey t ; p t; w t ; e! t 1 ] 0 ; z t [r n t v t ;! n t ] 0 To examine uniqueness of equilibrium, the relevant matrix is A w A 1 w;0 A w;1 There are three endogenous variables ey t ; p t; w t and one predetermined variable e! t 1. The system of di erence equations should thus have three unstable roots and one stable root This corresponds to A w having three characteristic roots within the unit circle, and one root outside the unit circle (or, had we written the system as E t fx t+1 g = A 1 w x t ::::, then the Blanchard and Kahn criterion would state that there should be three roots outside the unit circle; cf. the note on web) No analytical solution is available, but numerical analysis indicates that p + w > 1 is a su cient condition for equilibrium determinacy a generalized Taylor principle Immediate policy insight: Closing all gaps and having zero in ation rates are not generally feasible: Even if rt n = v t, ey t = p t = w t in real wages So, only if! n t = 0, is a gapless equilibrium possible = 0 will not be possible, as productivity shocks will require changes 19

21 Dynamic responses following a contractionary monetary policy shock (with simple Taylor rule based only on price in ation to facilitate comparison with Chapter 3): 20

22 Welfare-relevant objective for monetary policy Given that all gaps generally cannot be closed, what are the objectives of monetary policy? Social planner problem: s.t. (1), (2) and (6) Solution is max Z 1 0 U (C t (j) ; N t (j)) dj, all t C t (i; j) = C t ; all i; j 2 [0; 1] N t (i; j) = N t (j) = N t (i) = N t ; all i; j 2 [0; 1] U n;t = MP N t U c;t Note with monopoly price and wage setting under exible prices, where is an employment subsidy W t P t = U n;t U c;t M w ; P t = M p (1 ) W t MP N t Let 1 = (M w M p ) 1 then U n;t U c;t = MP N t and the ex-price allocation is e cient. This is assumed in the remainder 21

23 By the same method of deriving the approximated welfare under price rigidities only, the welfare loss can be written as a second-order approximation under both price and wage rigidities: W = 1 2 E 0 t + ' + ey t 2 + " p ( p 1 t) 2 + " w (1 ) ( w t ) 2 + t.i.p. (25) p w t=0 (Measured as a fraction of steady-state consumption.) Hence, a trade-o is present between output gap stability, price in ation stability, and wage in ation stability The relative weights on each objective have immediate intuition: Price in ation is more costly if the elasticity of substitution between goods are higher (as consumption dispersion will be large) Wage in ation is more costly if the elasticity of substitution between labor types are higher (as labor dispersion will be large) More price (wage) rigidity makes price (wage) in ation more costly as it rises price (wage) dispersion Note how the special case of exible wages ( w! 1) makes the wage in ation term vanish (and vice versa for the price in ation term in the special case of exible prices) 22

24 Optimal commitment policy Optimal monetary policy is computed under the assumption that commitment is possible (to derive a benchmark against which simple rules are to be compared) Following the approach of Chapter 5, one nds the optimal sequences of ey t, p t, w t, e! t to minimize W subject to the two Phillips curves and the dynamic de nition of the real wage gap From the four rst-order conditions and the three equilibrium condition, one solves for the endogenous variables and the three Lagrange multipliers (we need not set up the system here) Only in a special case can an analytical solution be attained: When substitution elasticities are proportional and the output gap a ects wage and price in ation identically: The optimal policy involves " p = " w (1 ), and p = w t = 0; which always implies ey t = 0 irrespective of parameters where t w p t + p w t p + w p + w I.e., a weighted average of in ation rates are fully stabilized; highest weight to price with highest degree of nominal rigidity (lowest z, z = p; w) 23

25 Dynamic responses following a positive technology shock: Optimal policies (commitment): General case 24

26 The performance of simple interest-rate rules Following the optimal commitment policies may be complicated Simple rules could be a substitute, and their performance are analyzed numerically Three strict in ation targeting rules: p t = 0, or w t = 0, or t = 0 (which implies ey t = 0) Three Taylor-type rules, called exible rules of the following kind i t = + 1:5 p t; i t = + 1:5 w t ; i t = + 1:5 t ; In each case, the model is numerically solved, and the s.d.s of the three welfare-relevant macrovariables are listed along with the welfare losses 25

27 26

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