1 The model
Consider the following model.
Households allocate their resources between consumption Ct, invest-
ments It and government-issued bonds Bt. They receive income from labor services Wtht, from proÖts Dt, from renting capital services Kt at the rate Rtk and from holding government bonds.
Householdsí utility function is:
X1 “C1″# h1+’ #
E0 *t t !t (1)
t=0 1!+ 1+’ Householdsí budget constraint (in nominal terms) is:
PtCt +PtIt +Bt =Rt”1Bt”1 +Wtht +Dt +RtkKt”1 !PtTt (2) Capital accumulation equation:
Kt = (1 ! /) Kt”1 + It (3)
Pt is the consumption price index Rt is the (gross) nominal interest rate, Kt is the physical capital stock and Tt are lump-sum taxes.
Labor decisions are made by a central authority within the house- hold, a union, which supplies labor monopolistically to a continuum of labor markets. Wt is an index of nominal wages prevailing in the econ- omy. In each particular labor market, the union takes Wt as exogenous. The union is assumed to supply enough labor to satisfy demand. Each agent provides each possible type of labor input. This assumption avoid heterogeneity across households in the number of hours worked. This speciÖcation gives rise to a wage-ináation Phillips curve with a larger coe¢cient on the wage-markup gap that the model with employment heterogeneity across households.
We assume that each period only a fraction 1 ! 0w of unions, drawn randomly from the population, reoptimize their nominal wage. All unions resetting their wage in any given period will choose the same wage, since they face an identical problem.
The Önal good Yt is produced under perfect competition. Intermedi- ate Örms are monopolistically competitive and use as inputs capital and labor services, Kt and ht respectively. The production technology is:
Y=K& h1″& t t”1t
Firms maximize their (nominal) proÖts Dt = PtYt ! Wtht ! RtkKt”1 subject to the technology constraint, choosing how much inputs Kt and ht to use.
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Following Calvo (1983), each Örm may reset its price only with prob- ability 1!0p in any given period, independently of the time elapsed since the last adjustment. Thus, each period a measure 1 ! 0p of producers reset their prices, while a fraction keep their prices unchanged. As a result, the average duration of a price is given by $1 ! 0p%”1. In this context, 0p becomes a natural index of price stickiness.
The government budget constraint is:
Rt”1Bt”1 = Bt + PtTt
The monetary authority sets the nominal interest rate according to
a Taylor rule.
2 First order conditions
The Örst order conditions obtained from this model are the following. From the householdsí problem, we obtain:
C”# =2 (4) tt
R =3p 2t (5) t t+1 *2t+1
2t+1 * &rk + (1 ! /)’ = 1 (6) 2t t+1
Kt = (1 ! /) Kt”1 + It (7) M R S t ” ! U h ( C t ; h t ) = C t # h ‘t = h ‘t (8)
UC (Ct;ht) 2t The following 4 equations deÖne wage dynamics:
( 1 )1″”w (Wt#)1″”w
1=0w 3w +(1!0w) W (9)
tt W#=”w wt
t ” w ! 1 ; wt
wt = 2tMRStht + *0wEt
;w = 2 ht + *0 E ;w
t tPt w t t+1
From the Örmsí problem, we obtain:
Kt”1 = < wt h t ( 1 ! < ) r tk
(10) wt+1 (11)
(12)
(13)
2
mc = <“& (1 ! <)”(1″&) $rk%& w1″& (14) ttt
Y=K& h1″& (15) t t”1t
The following 4 equations deÖne price dynamics:
Taylor rule:
( 1 )1″”p $ %(Pt#)1″”p
3p + 1!0p P (16)
tt
P# ” p
t=pt (17)
1=0p
Market clearing:
Yt = Ct + It (20) Rt (Rt”1 )+R “(3pt )+$ (Yt )+y #1″+R
R=R 3p Y (21) Wage ináation deÖnition:
3wt = wt 3pt (22) wt”1
where 2t is the Lagrange multiplier for the problem of households (real terms budget constraint), 3wt is gross wage ináation, 3pt is gross price ináation, wt is the real wage and rtk is the real price of capital. “w is the elasticity of substitution between labour services and “p is the elasticity of substitution between goods.
3 Steady state
The steady state of the model is derived as follows:
3p = 1 R = 3p
*
rk = 1 ! (1 ! /) *
P t ” p ! 1 ; pt
p = 2 Y mc + *0 E 3″p p (18)
t t t t p t t+1 t+1
;p =2Y +*0 E3″p”1;p (19)
t tt ptt+1t+1
3
I=/ K
mc = “p ! 1 “p
Y = rk 1 K <mc
II =K
YY K
C=1!I YY
P# =P “#1
“p !1<& (1!<)1″& 1!& “p (rk)&
w=
K=<w
h 1!<rk
Y =(K)”(1″&)
Kh
M RS = w “w ! 1
“w
C YK K (K)& K h=Kh!/h= h !/h
“()#1 h= MRS C “# ‘+’
h
The set of log-linearized equations is the following:
! + C^ t = 2^ t
2^t = 2^t+1 + R^t ! 3^pt+1
4 Log-linearized model
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^^ 2 =2
k
w w (1!0w)(1!*0w), \- 3^t =*3^t+1! 0 w^t!MRSt
w
+[1!*(1!/)]r^
t t+1 t+1
K^=(1!/)K^ +/I^ t t”1t
\^^ MRSt = +Ct + ‘ht
^^
K !N=w^!r^
t”1 t t t k
mcc =<r^ +(1!<)w^ ttt
Y^ t = < K^ t ” 1 + ( 1 ! < ) h^ t
3^pt =*3^pt+1+$1!0p%$1!*0p%mcct
k
0p C C^ + I I^ = Y^
YtYtt
R^ t = ; R R^ t ” 1 + ( 1 ! ; R ) , ; 0 3^ pt + ; y Y^ t – 3^wt = w^t ! w^t”1 + 3^pt
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