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

\begin{array}{l} ω_{j, t} = \frac{1}{α_{j}^{α_{j}} {(1 - α_{j})}^{1 - α_{j}}} \frac{1}{{(z_{j, t})}^{α_{j}}} \frac{{\tilde{w}}_{t}^{α_{j}}}{r_{j, t}} \\ \begin{matrix} l_{j, t}^{d} = \frac{α_{j} ω_{j, t}}{{\tilde{w}}_{t}} {\tilde{y}}_{j, t} & j = 1, ..., N \\ {\tilde{y}}_{j, t} = γ_{j} {(r_{j, t}^{+})}^{- ε} {\tilde{y}}_{t} & j = 1, ..., N \end{matrix} \\ \begin{matrix} {\tilde{m}}_{j, t}^{d} \end{matrix} = (1 - α_{j}) ω_{j, t} r_{j, t} {\tilde{y}}_{j, t} . \end{array}