The Basic New Keynesian Model, the Labor Market and Sticky Wages

The Basic New Keynesian Model, the Labor Market and Sticky Wages Lawrence J. Christiano August 25, 203

Baseline NK model with no capital and with a competitive labor market. private sector equilibrium conditions Details: http://faculty.wcas.northwestern.edu/~lchrist/course/korea_202/intro_nk.pdf Standard Labor Market Friction: Erceg-Henderson-Levin sticky wages.

New Keynesian Model with Competitive Labor Market: Households Problem: max E 0 Â b t t=0! log C t exp (t t ) N+j t, t t = lt + t + # t t j s.t. P t C t + B t+ W t N t + R t B t + Profits net of taxes t First order conditions: = be t C t C t+ exp (t t ) C t N j t = W t P t. R t p t+ (5)

New Keynesian Model with Competitive Labor Market: Goods Final good firms: maximize profits: subject to: Z P t Y t P i,t Y i,t dj, 0 Z Y t = 0 # Y # # # i,t dj. Foncs: # Pt Y i,t = Y t! P i,t "cross price restrictions" Z z } { P t = 0 P (#) i,t di #

New Keynesian Model with Competitive Labor Market: Goods Demand curve for i th monopolist: # Pt Y i,t = Y t. P i,t Production function: Y i,t = exp (a t ) N i,t, a t = ra t + # a t Calvo Price-Setting Friction: P i,t = P t with probability q with probability q Real marginal cost: s t = dcost dworker doutput dworker = P i,t minimize monopoly distortion by setting = # # z } { ( n) W t P t exp (a t ).

Brief Digression With Dixit-Stiglitz final good production function, there is an optimal allocation of resources to all the intermediate activities, Y i,t It is optimal to run them all at the same rate, i.e., Y i,t = Y j,t for all i, j 2 [0, ]. For given N t, it is optimal to set N i,t = N j,t for all i, j 2 [0, ] In this case, final output is given by Y t = e a t N t. Best way to see this is to suppose that labor is not allocated equally to all activities. But, this can happen in a million di erent ways when there is a continuum of inputs! Explore one simple deviation from N i,t = N j,t for all i, j 2 [0, ].

SupposeLaborNot AllocatedEqually Example: N it 2N t i 0, 2 2 N t i,,0. 2 Notethatthisisaparticulardistributionof laboracrossactivities: 0 Nit di 2 2N t 2 2 N t N t

LaborNot AllocatedEqually,cnt d Y t Yi,t di 0 0 2 Y i,t e at 0 2 di 2 N i,t Yi,t di 2 di Ni,t di e at 2 2N t di 0 2 Nt 2 e at 2 N t 2 di 0 2 2 e at N t e at 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

Let Optimal Price Setting p t P t P t, p t P t P t. Optimal price setting: where p t = K t F t, K t = # # s t + bqe t p t+ # K t+() F t = + bqe t p t+ # t+.(2) Note: K t = # # s t + bqe t p t+ # # # s t+ + (bq) 2 E t p t+2 # # # s t+2 +...

Goods and Price Equilibrium Conditions Cross-price restrictions imply, given the Calvo price-stickiness: h i P t = ( q) P (#) t + qp (#) # t. Dividing latter by P t and solving: " # q p # # t p t =! K t = q F t " # q p t # # q (3) Relationship between aggregate output and aggregate inputs: C t = pt A t N t, (6) 2 where p t = 4( q) q p (#) t q! # # + q p# t p t 3 5 (4)

Linearizing around E cient Steady State In steady state (assuming p =, n = # # ) p =, K = F = bq, s = #, Da = t = 0, N = # Linearizing the Tack Yun distortion, (4), about steady state: ˆp t = q ˆp t! ˆp t = 0, t large Denote the output gap in ratio form by X t : X t A t exp C t so that (using x t ˆX t ): = p t t N t exp t +j x t = ˆN t + dt t +j tt, + j

NK IS Curve, Baseline Model The intertemporal Euler equation, (5), after substituting for C t in terms of X t : where X t A t exp then, use t t +j = be t X t+ A t+ exp = E t X t X t+ Rt+ R t p t+, t t+ +j Rt+ b exp a t+ a t t t+ t t + j dz t u t = ẑ t + û t, \ ut = û t ẑ t z t R t p t+

NK IS Curve, Baseline Model The intertemporal Euler equation, (5), after substituting for C t in terms of X t : where X t A t exp then, use to obtain: t t +j = be t X t+ A t+ exp = E t X t X t+ Rt+ R t p t+, t t+ +j Rt+ b exp a t+ a t t t+ t t + j dz t u t = ẑ t + û t, \ ut = û t ẑ t ˆX t = E t ˆX t+ ˆR t b p t+ ˆR t+ z t R t p t+

Let: NK IS Curve, Baseline Model R t exp (r t )! ˆR t = d exp (r t) exp (r) = exp (r) dr t exp (r) = dr t r t r. Also, Rt+ = exp log Rt+! ˆR t+ = d exp log Rt+ exp (log R = d log Rt+ ) =r in e cient steady state, with p= = log Rt+ z } { log R So, (letting r t E t log R t+ ) E t ˆR t ˆR t+ = drt E t d log R t+ = r t r t.

NK IS Curve, Baseline Model Substituting E t ˆR t ˆR t+ = rt rt into ˆX t = E t ˆX t+ ˆR t b p t+ ˆR t+, xt ˆX t, and using b p t+ = p t+, when p =, we obtain NK IS curve: x t = E t x t+ E t [r t p t+ rt ] Also, rt = log (b) + E t a t+ a t t t+ t t. + j

Linearized Marginal Cost in Baseline Model Marginal cost (using da t = a t, dt t = t t because a = t = 0): s t = ( n) w t, w t = exp (t t ) N j t A C t t! b w t = t t + a t + ( + j) ˆN t Then, ŝ t = b w t a t = (j + ) h tt j+ + ˆN t i = (j + ) x t

Linearized Phillips Curve in Baseline Model Log-linearize equilibrium conditions, ()-(3), around steady state: Substitute (3) into () ˆF t + ˆK t = ( bq) ŝ t + bq #b p t+ + ˆK t+ () ˆF t = bq (# ) b p t+ + bq ˆF t+ (2) ˆK t = ˆF t + q q b p t (3) q q ˆp t = ( bq) ŝ t + bq #b p t+ + ˆF t+ + q q ˆp t+ Simplify the latter using (2), to obtain the NK Phillips curve: p t = (q)(bq) q ŝ t + bp t+

The Linearized Private Sector Equilibrium Conditions of the Competitive Labor Market Model x t = x t+ [r t p t+ rt ] p t = (q)(bq) q ŝ t + bp t+ ŝ t = (j + ) x t h i rt = log (b) + E t a t+ a t t t+t t +j

EquationsofActualEquilibrium ClosedbyAddingPolicyRule E t t x t t 0(Phillipscurve) r t E t t r t E t x t x t 0(ISequation) r t t x x t r t 0(policyrule) r t a t t 0(definitionofnaturalrate)

SolvingtheModel s t a t t 0 0 a t t t t s t Ps t t 0 0 0 t 0 0 t 0 0 0 0 0 0 x t r t 0 x 0 x t r t 0 0 0 0 r t 0 0 0 rr t 0 0 0 0 t 0 0 0 0 0 0 0 0 0 0 0 0 x t r t 0 0 0 0 0 0 s t 0 0 0 0 s t 0 0 0 0 r t 0 0 0 E t 0 z t z t 2 z t 0 s t s t 0

SolvingtheModel E t 0 z t z t 2 z t 0 s t s t 0 Solution: Asbefore: 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

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 0.03 0.02 0.5 0. 0.5 0. natural nominal rate actual nominal rate 0.0 0.05 0.05 0 2 4 6 natural real rate 0.2 0.5 0. 0.05 0 2 4 6 employment 0.2 0.5 0. 0.05 0 0 5 0 0.5 0 2 4 6 log, technology 0 0 2 4 6 natural employment actual employment.2.5..05 0 2 4 6 output natural output actual output 0 2 4 6

Dynamic Response to a Preference Shock 0. 0.08 0.06 0.04 0.02 inflation 0.25 0.2 0.5 0. 0.05 output gap 0.25 0.2 0.5 0. 0.05 nominal rate natural real rate actual nominal rate 0 2 4 6 natural real rate 0.25 0 2 4 6 preference shock 0 2 4 6 output 0.2-0. 0.5 0. 0.05 0.5-0.2-0.3-0.4 natural output actual output 0 2 4 6 employment 0 0 2 4 6-0.5 0 2 4 6-0. -0.2-0.3-0.4-0.5 0 2 4 6 natural employment actual employment

Reasons to consider frictions in the labor market: Play an essential role in accounting for response to a monetary policy shock. With flexible wages, wage costs rise too fast in the wake of expansionary monetary policy shock. High costs limit firms incentive to expand employment. High costs imply sharp rise in inflation. But, the data suggest that after an expansionary monetary policy shock inflation hardly rises and output rises a lot! Wage frictions play essential role in making possible an account of monetary non-neutrality (see CEE, 2005JPE). Important for understanding employment response to other shocks too. Introducing sticky wages and monopoly power in labor market provides a theory of unemployment.

Sticky Wages Basic model is due to Erceg-Henderson-Levin. We will follow the interpretation of EHL suggested by Gali, so that we have a theory of unemployment (see also Gali-Smets-Wouters). We will not go into the details here see Christiano-Trabant-Walentin Handbook chapter. also, detailed online lecture notes (links in course syllabus).

Outline Provide a broad sketch of the model. Discuss some linearized equations of the model. Provide a critical assessment of the model.

Ba Model There are many identical households. The representative household contains all workers: Many workers of each type. Differentiated by utility cost of working Every worker enjoys the same level of consumption.

Household A plumbers painters etc. Monopoly plumberunion Household B plumbers painters Monopoly painterunion etc.

Typej MonopolyUnion Wage unemployment Labor supply onopoly age Labor demand markup #of people employed Labor force Quantity of labor

Does(the(Degree(of(Union(Power( Affect(the(Unemployment(Rate?( OECD(Employment(Outlook((2006,(chap(7)( Norway(and(Denmark(have(unioniza8on(rates( near(80(percent.(before(the(current(crisis(their( unemployment(rate(was(under(3.0(percent.(

Union(Density(Rates( Jelle(Visser,(2006(Monthly(Labor(Review( Union(density(rates,(970,(980(and(990 2003,( adjusted(for(comparability.( Defini8on:(union(membership(as(a(propor8on(of( wage(and(salary(earners(in(employment.(

Unemployment(Rates:(Sources( BLS,(Interna8onal(Comparisons(of(Annual( Labor(Force(Sta8s8cs,(Adjusted(to(U.S.( Concepts,(0(Countries,(970-200,(Table(-2.( Finland,(Norway(and(Spain(taken(from(ILO,( Comparable(annual(employment(and( unemployment(es8mates,(adjusted(averages(

Monopoly(Power(Hypothesis( If(union(density(in(country(A(grows(faster(than( union(density(in(us,(then( Expect(unemployment(in(country(A(to(rise(more( than(unemployment(in(us.(( Test(is(based(on(low(frequency(part(of(the( data,(not(on(the(levels.(

Wage Setting with Frictions Suppose, for simplicity, that there are no shocks to labor preferences. The solution to union problem gives rise to a wage Phillips curve : 0 ˆp w,t = = [ C t N j t k z } { w BĈ + j ˆN t b w t C + j# w @ A + b ˆp w,t+ k w = ( q w)( bq w ) q w, p w,t W t W t, w t W t P t.

Collecting the Equations Variables to be determined: x t, r t, p t, ˆr t, ŝ t, b w t, ˆp w,t Equations: x t = x t+ [r t p t+ rt ] p t = (q)(bq) q ŝ t + bp t+ ŝ t = b w t a t ˆp w,t = k w +j# w Ĉt + j ˆN t b w t + b ˆpw,t+ rt = log (b) + E t [a t+ a t ] Need more equations: relate Ĉ t, ˆN t to x t and shocks, and connect b w t to ˆp w,t and p t.

Recall definition of level of output gap: X t : X t C t A t = p t N t. Then, in the linear approximation about undistored steady state, z} { Ĉ t = x t + a t, ˆN t =! Ĉ + j ˆN t = ( + j) x t + a t. Also, p w,t W t W t = W t P t W t P t P t P t =! ˆp w,t = b w t b w t + b p t w t w t p t

Equations of the Sticky Wage Model Six private sector equations in seven variables: x t = x t+ [r t p t+ rt ] p t = (q)(bq) q ŝ t + bp t+ ŝ t = b w t a t ˆp w,t = k w +j# w ( + j) xt + a t b w t + b ˆpw,t+ ˆp w,t = b w t b w t + ˆp t r t = Da t+ where r t and r t now stand for their deviations from steady state, r. Monetary policy rule: r t = ar t + ( a)[f p p t + f x x t ] + u t, where u t denotes a monetary policy shock.

Conclusion Sticky wages e ective in getting wage not to rise much after a monetary policy shock, limiting the rise in inflation and amplifying the rise in output. But, Underlying monopoly power theory of unemployment does not receive strong support in the data. Important recent policy question: what will be the e ect of extending the duration of unemployment benefits? some say, more unemployment benefits will make an already weak economy even weaker. others say, the number of available jobs is low because of weakness in aggregate demand. if workers search less in response to better unemployment benefits, that won t change the (low) number of existing jobs because it won t a ect aggregate demand (see Christiano, Eichenbaum and Trabandt (in process)). We will discuss alternative approaches to labor markets next (see Christiano, Eichenbaum and Trabandt (203)).