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

\begin{array}{l} r_{j, t} {(k)}^{*} = \frac{ε}{ε - 1} ω_{j, t} \\ \Rightarrow 1 = \frac{ε}{ε - 1} ω_{j, t} \\ \Rightarrow r_{j, t} = \frac{ε}{ε - 1} \frac{1}{α_{j}^{α_{j}} {(1 - α_{j})}^{1 - α_{j}}} {(\frac{{\tilde{w}}_{t}}{z_{j, t}})}^{α_{j}} \\ r_{j, t} = r_{j, t - 1} \frac{Π_{j, t}}{Π_{t}} . \end{array}