Unemployment and Business Cycles — A Simple Macro Model —

Lawrence J. Christiano Martin Eichenbaum Mathias Trabandt

2nd Annual IAAE Conference, University of Macedonia, Thessaloniki, Greece 2015

A Simple Macro Model ... 2

! Competitive Önal goods production: Yt = 4

Z1 0

! jth input produced by monopolistic ëretailersí:

1 lf

3 lf

Yj,t dj5 .

ñ Production: Yjt = exp(at )hj,t .

at = tat"1 + #t ñ Homogenous good, hj,t , purchased in competitive markets for real price, Jt . ñ Retailersí prices subject to Calvo sticky price frictions (no price indexation). ! Homogeneous input good ht produced by the Örms in our AOB

labor market model.

A Simple Macro Model ...

! Representative household: •

E0 Â bt ln Ct t=0

Pt Ct + Bt+1 " Wt lt + Pt D (1 # lt ) + Rt#1 Bt + Tt ! Central bank follows Taylor rule.

Calvo Sticky Wages ! Erceg, Henderson and Levin (2000, JME). ! Retailer (subject to Calvo sticky prices) purchases homogenous

labor hj,t from representative labor contractor. ! Representative contractor produces homogeneous labor input by combining di§erentiated labor inputs, li,t , i 2 (0, 1) : !Z 1 # lw 1 l ht = , lw > 1. (li,t ) w di 0

! Labor contractors are perfectly competitive and take the

nominal wage rate, Wt , of ht and Wi,t , of the ith labor type as given. ! ProÖt maximization on the part of contractors implies: li,t =

$

Wt Wi,t

%

lw lw #1

ht .

(1)

Calvo Sticky Wages... ! Continuum of households indexed by i 2 (0, 1) .

! The ith household is the monopoly supplier of li,t and chooses

Wi,t subject to (1) and Calvo wage-setting frictions.

ñ With probability 1 # x w optimizes the wage, Wi,t . With probability x w wage rate given by: Wi,t = Wi,t#1 . ! Households insured against the idiosyncratic uncertainty

associated with the Calvo wage-setting friction ! Household preferences: 1+ y

ln Ct # {

li,t

1+y

, { > 0, y $ 0.

(2)

Calvo Sticky Wages...

! Optimization problem of the ith household: •

maxE0 Â ( bx w )i ˜t W

i=0

subject to: lt+i =

#

"

1+ y # lt+i ˜ t lt+i " { u t+i W . 1+y

Wt+i ˜t W

$

lw lw "1

ht + i

(3)

1. Simple Macro Model Based on ìUnemployment and Business Cyclesî by Christiano, Eichenbaum and Trabandt (2015). Here, we list the dynamic equilibrium equations for the simple macro model with alternating o§er bargaining and hiring costs. We also provide steady state calculations. For completeness, we also list the equilibrium equations for alternative labor market speciÖcations in the simple macro model such as: i) search costs instead of hiring costs, ii) Nash sharing instead of alternating o§er bargaining and iii) sticky wages as in Erceg, Henderson and Levin (2000). The setup of these alternative labor market speciÖcations is the same as in the medium-sized DSGE model.

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1.1. Simple Macro Model: Dynamic Equations Pricing 1 (1) : Kt = #t = exp(at ) + %&Et ( "t+1 Kt+1 where ) = *=(* ! 1) Pricing 2 (2) : Ft = 1 + %&Et ( "!1 t+1 Ft+1

Pricing 3 (3) : &( "!1 = 1 ! (1 ! &) (Kt =Ft )1!" t !" # $"=("!1) Price dispersion (4) : 1=p"t = (1 ! &) 1 ! -( "!1 = (1 ! &) + &( "t =p"t!1 t

Resource constraint (5) : p"t exp(at )lt = Ct + 0Q$!1 t xt lt!1 where 3 2 f0; 1g Euler equation (6) : 1 = %Et Ct =Ct+1 Rt =( t+1 p Present value wages (7) : wtp = wt + 7%Et Ct =Ct+1 wt+1

PV marg. revenue (8) : #pt = #t + 7%Et Ct =Ct+1 #pt+1 Zero proÖt/free entry (9) : Jt = 0Q$!1 t Value of Örm (10) : Jt = #pt ! wtp

Work value (11) : Vt = wtp + At

Contin. work value (12) : At = (1 ! 7) %Et Ct =Ct+1 [ft+1 Vt+1 + (1 ! ft+1 ) Ut+1 ] +7%Et Ct =Ct+1 At+1 Unempl. value (13) : Ut = D + %Et Ct =Ct+1 [ft+1 Vt+1 + (1 ! ft+1 )Ut+1 ] Alt. O§er. Sharing (14) : >1 Jt = >2 (Vt ! Ut ) ! >3 ? + >4 (#t ! D) Job Önding rate (15) : ft = xt lt!1 =(1 ! 7lt!1 ) Vacancy Ölling rate (16) : Qt = ft =-t Matching function (17) : ft = @ m -1!' t Lab. mkít tightness (18) : -t = (vt lt!1 ) =(1 ! 7lt!1 ) LOM empl. (19) : lt = (7 + xt ) lt!1 Unempl. rate (20) : ut = 1 ! lt Taylor rule (21):

ln (Rt =R) = 7R ln (Rt!1 =R) + (1 ! 7R ) [r) ln (( t =() + ry ln (lt =l)] + "R;t

Technology (22) : at = E at!1 + "t Check: 22 equations in the following 22 variables: at Rt ut lt ( t xt -t vt ft Qt Ut Ct Vt wt Jt #t p"t Kt Ft At #pt wtp Note that 3 = 1 corresponds to the hiring cost speciÖcation while 3 = 0 corresponds to the search cost speciÖcation. Also, the case of Nash sharing can be obtained by replacing the alternating o§er sharing rule (14) with the following equation: 3

Nash Sharing (14í): Vt ! Ut = F [Vt ! Ut + Jt ] 1.2. Simple Macro Model: Steady State IMPOSE ( = 1; ìdropî equation (21), i.e. R = R (22)

:

a=0

(1-3)

:

#=1

(1)

:

K = #=(1 ! %&)

(3)

:

F =K

(4)

:

p" = 1

(6)

:

R = 1=%

(19)

:

x=1!7

(20)

:

u = 1 ! l ! assume l; solve (14) or (14í) for ? or F later

(5)

: :

C = (1 ! sl ) p" ea l

where sl = 0Q$!1 xl=(p" ea l)

! 0 = (sl p" ea l) =(Q$!1 xl) (15)

:

f = xl=(1 ! 7l)

(16)

:

- = f =Q ! assume Q; solve (17) for @ m

(17)

:

@ m = f =-1!'

(18)

:

v = -(1 ! 7l)=l

(9)

:

J = 0Q$!1

(8)

:

#p = #=(1 ! 7%)

(10)

:

(7)

:

(11-13)

: :

(13)

:

(11)

:

(14)

:

(14í)

:

w p = #p ! J

w = wp (1 ! 7%) % V ! U = wp !

D 1 ! %7 where D = D=w & w D + %f (V ! U ) U= 1!% A = V ! wp

& % & 1 ! (1 ! f ) %7 = 1 ! %7

? = (>2 (V ! U ) + >4 (# ! D) ! >1 J) =>3 V !U F= V !U +J 4

1.3. Simple Macro Sticky Wage Model: Dynamic Equations Price setting 1 (1) : Kt = #t = exp(at ) + %&Et ( "t+1 Kt+1 where ) = *=(* ! 1) Price setting 2 (2) : Ft = 1 + %&Et ( "!1 t+1 Ft+1

Price setting 3 (3) : &( "!1 = 1 ! (1 ! &) (Kt =Ft )1!" t !" # $"=("!1) Price dispersion (4) : 1=p"t = (1 ! &) 1 ! &( "!1 = (1 ! &) + &( "t =p"t!1 t !" # $"w =("w !1) " w !1 w Wage dispersion (5) : 1=wt" = (1 ! & w ) 1 ! & w ( "w;t =(1 ! & w ) + & w ( "w;t =wt!1

Wage setting 1 (6) : Kw;t = (wt" lt )1+ + %& w Et (( w;t+1 )"w (1+ ) Kw;t+1 where ) = *w =(*w ! 1) w !1 Wage setting 2 (7) : Fw;t = (*w ! 1)=*w wt" lt =Ct + %& w Et (wt+1 =wt ) ( "w;t+1 Fw;t+1 w !1 Wage setting 3 (8) : & w ( "w;t = 1 ! (1 ! & w ) ({Kw;t = (Fw;t wt ))(1!"w )=(1!"w +(1+

)"w )

Wage ináation (9) : ( w;t = wt =wt!1 ( t Resources (10) : p"t wt" exp(at )lt = Ct Euler equation (11) : 1 = %Et Ct =Ct+1 Rt =( t+1 Real wage (12) : #t = wt Taylor rule (13) : ln (Rt =R) = 7R log (Rt!1 =R) + (1 ! 7R ) [r) ln (( t =() + ry ln (lt =l)] + "R;t Technology (14) : at = E at!1 + "t Check: 14 equations in the following 14 variables, at Rt lt ( t Ct wt #t p"t Kt Ft wt" Kw;t Fw;t 2w;t Notice that linearizing equations (1)-(3) and (6)-(8) results in the usual Phillips curves, h i ^ t ! at ^ t = %Et 2 ^ t+1 + (1 ! &) (1 ! %&) # 2 & h i ^ w;t = %Et 2 ^ w;t+1 + (1 ! & w ) (1 ! %& w ) ^lt + C^t ! w^t 2 & w (1 + *w ) Also, & w = 0 and *w ! 1; implies that the non-linear equations (1)-(14) reduce to the standard Clarida-Gali-Gertler (1999) model.

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1.4. Simple Macro Sticky Wage Model: Steady State IMPOSE ( = 1; ìdropî equation (13), i.e. R = R (14) : a = 0 (1) : # = 1 (4) : p" = 1 (5) : w" = 1 (9) : ( w = 1 (11) : R = 1=% (12) : w = # (2) : F = 1=(1 ! %&) (3) : K = F (10) : C = w" p" ea l ! assume l, solve (8) for A (6) : Kw = l1+ =(1 ! %& w )

(7) : Fw = (*w ! 1)=*w l=C=(1 ! %& w ) " # (8) : A = Fw =Kw = (*w ! 1)=*w = Cl

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