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7.5 Derivation of the New Keynesian Phillips Curve
Define MCt ≡ Wt
MP Lt and MP Lt ≡ (1 − α) Yt
Lt
for real marginal costs and the marginal product of labor,
respectively. Rewrite the price-setting equation of the firm in terms of real marginal costs
νMCt − (ν − 1) = φR
� Πt
Πss
� � �
Πt
Πt+1
− 1 − Et Mt+1φR − 1
Πss
Πss
Log-linearizing the equation above around the nonstochastic steady-state gives
where γ1 = ν−1
state. 34
φR , γ2 = β∆Y
1 1− ψ
ss
�πt = γ1 �mct + γ2Et[�πt+1]
� ∆Yt+1Πt+1
, and lowercase variables with a tilde denote log deviations from the steady-
Substituting in the expression for the marginal product of labor and imposing the symmetric equilibrium
conditions, real marginal costs can be expressed as
MCt =
WtLt
(1 − α)K α t (AtNtLt) 1−α)
Define the following stationary variables: W t ≡ Wt
Kt and N t ≡ Nt
. Thus, we can rewrite the expression
above as
Log-linearizing this expression yields
MCt =
W tL α t
(1 − α)(AtN t) 1−α
Kt
�mct = �wt + α � lt − (1 − α)�at − (1 − α)�nt
where lowercase variables with a tilde denote log deviations from the steady-state.
34 In a log-linear approximation, the relationship between the price adjustment cost parameter φR and the fraction
of firms that resetting prices (1 − θc) is given by: φR =
(ν−1)θc
(1−θc)(1−βθc)
36
Πss
�