RDP 2010-01: Reconciling Microeconomic and Macroeconomic Estimates of Price Stickiness Equation

\begin{array}{l} l_{t}^{\frac{1}{η}} = \frac{{\tilde{w}}_{t} a_{t}}{{\tilde{c}}_{t}} \\ \frac{a_{t}}{{\tilde{c}}_{t}} = β E_{t} (\frac{a_{t + 1}}{{\tilde{c}}_{t + 1}} \frac{1}{Π_{t + 1}} I_{t} e^{μ_{z} + ε_{z, t + 1}}) \\ \frac{{\tilde{c}}_{t}}{a_{t} {\tilde{h}}_{t}} = 1 - \frac{1}{I_{t}} \\ {\tilde{Λ}}_{t} = \frac{a_{t}}{{\tilde{c}}_{t}} \\ {\tilde{h}}_{t} = Ξ_{t} {\tilde{h}}_{t - 1} \frac{1}{Π_{t}} e^{- μ_{z} - ε_{z, t}} \\ g_{t} = \frac{{\tilde{c}}_{t}}{{\tilde{c}}_{t - 1}} e^{- μ_{z} - ε_{z, t}} \\ I_{t} = I_{t - 1}^{ρ_{i}} {(\frac{1}{β} e^{- (1 - ϕ_{g}) μ_{z}} Π g_{t}^{ϕ_{g}})}^{1 - ρ_{i}} e^{ε_{i, t}} . \end{array}