(5)
where we define ms,j ≡ M-^.
The necessary conditions for the households with respect to consumption cts,j , s =
1, . . . , T + T R , next-period capital kts++11,j , and next-period money mts++11,j at age s =
1, . . . , T + T R - 1 in period t are as follows:
s,j
t
Mts,j
uc Γ , 1
nj = γ ( cSj )γ (1σ) ) -1 f
sj (1-γ)(1-σ )
ms,j
πt
s,j
t
s,j
t
βEt
βE
ΛS+11j (1 - δ + r,+1 (1 - τ0 y+1j—±1) ππt±ι))
si S+1+1 ,j m+ ,j 1 ns +1 ,jʌ
λs+1 ,j uM∕P ^ct +1 , Pt+1 , 1 - nt+1 J
πt+1 πt+1
= βEt
∖++1 ,j ∩ √∣ (rs+1 ,jAγ(1 σ) (ms+1 ,j)(1 -γ)(1 ~σ)-1
λt +1 + (1 - γ) kct+1 ) m++1 )____________
πt+1 πt+1
(6)
(7)
(8)
The optimal labor supply of the productivity j-type workers at age s = 1, . . . , T is
given by:
λts,jwte(s, j)
1 - τ0 (ys,j—´ — i = un c⅛, M^-, 1 - n j = η0 (1 - ns,j ) η ∙
π π Pt
(9)
2.2 Production
The description of the production sector is similar to Bernanke, Gertler, and Gilchrist (1999).
A continuum of perfectly competitive firms produce the final output using differentiated
intermediate goods distributed on [0,1]. These goods are manufactured by monopolis-
tically competitive firms. Firms in the intermediate goods’ sector set prices according
to Calvo (1983).
Final Goods Firms. The firms in the final goods sector produce the final good with
a constant returns to scale technology using the intermediate goods Yt(j), j ∈ [0, 1] as
an input:
Yt = (ʃ Yt ('j ) dj) ‘ 1 ∙ (10)
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