The general equilibrium effects of fiscal policy

where χ t ≡ λ t P t , which is of use also in log-linearization of (31)̂χ t = ̂R t + E t [̂χ t+1 ] − E t [̂π t+1 ] . (49)Defining π W t≡ W t /W t−1 one haŝπ W t = ̂ω t − ̂ω t−1 + ̂π t (50)and also, log-linearizing (32),[ ( )̂π W t =βE t [̂π t+1]+ W 1− τw(θφπ 2 L l p τ w)− l) ̂χ t −1− τ ŵτ w t +(θ ] L l p̂lp t − l̂l t + θ Ll σ l p [σ l̂lt +χφπ 2 ω̂l]p t−̂ω t +̂ε b t +̂ε l t(51)Defining q t = µ t (1+τ c t ) , log-linearization of (33) and (34) yieldsχ tÎ t =Ît−11 + β + β1 + β E t[Ît+1] +̂q ts ′′ (1 + β) −β1 + β E t[̂ε i t+1] + ̂εi t1 + β(52)andτ ĉχ t + ̂q t −1 + τ ̂τ c c t = E t [̂χ t+1 ]+ (53)+β[(1−δ)E t [̂q t+1 ] + r k (1+τ c )(1−τ k )E t [̂r t+1] k − r k τ k (1+τ c )E t [̂τ k t+1] − (1−δ)τ ]cE1+τ c t [̂τ c t+1]where we used the steady state equalities q = 1, s(.) = s ′ (.) = 0, u = 1, ψ(1) = 0,ψ ′ (1) = r k (1 − τ k ), and r k = [1 − β(1 − δ)]/β(1 − τ k )(1 + τ c ). Log-linearization of (35)directly follows:ψ ′′ (u)ψ ′ (u) ût = ̂r t k τ k−(1 − τ )̂τ k k t . (54)Zero steady state adjustment costs imply also that the log-linearized version of constraint(6) iŝkt+1 = (1 − δ) ̂k t + δÎt (55)wherêkt = û t + ̂k t , (56)and, from the capital market equilibrium (26),̂kt = ŷ t + (1 − α) [ ŵ t − ̂r k t − ẑ t]. (57)As for budget constraints, it is enough to log-linearize (7), the one of non-Ricardiansc NR ĉ NRt = ωl[(1 − τ w )(̂ω t + ̂l t ) − τ ŵτ w t ] + tr ̂tr t (58)37