Hello everyone, I have recently been replicating the GK-2011 paper and encountered a problem when handling the first-order conditions. I am unsure how to derive the following equations for price dispersion and optimal pricing. Thank you in advance!!!
Price Dispersion
D_t=\gamma D_{t-1} \Pi_{t-1}^{-\gamma_p \varepsilon} \Pi_t^{\varepsilon}+(1-\gamma)\left(\frac{1-\gamma \Pi_{t-1}^{\gamma_p(1-\gamma)} \Pi_t^{\gamma-1}}{1-\gamma}\right)^{\frac{-\varepsilon}{1-\gamma}}
Optimal Pricing
\Pi_t^*=\Pi_t \frac{\varepsilon}{\varepsilon-1} \frac{F_t}{Z_t}
Auxiliary Variable
\begin{gathered} F_t=P_t^m Y_t+E_t \Lambda_{t, t+1} \beta \gamma \Pi_{t+1}^{\varepsilon} \Pi_t^{-\varepsilon \gamma_p} F_{t+1} \\ Z_t=Y_t+E_t \Lambda_{t, t+1} \beta \gamma \Pi_{t+1}^{\varepsilon-1} \Pi_t^{\gamma_{p(1-\varepsilon)}} Z_{t+1} \end{gathered}
In fact, I once attempted to derive the price dispersion following Professor Jpfeifer’s method for deriving price dispersion of Gali(2015), but the results I obtained were different from those in the paper.
\begin{aligned}
D_t & =\int_0^1\left(\frac{P_t(i)}{P_t}\right)^{-\varepsilon} \, di \\
& =(1-\theta)\left(\frac{P_t^*}{P_t}\right)^{-\varepsilon}+\int_{D(i)}\left(\frac{P_{t-1}(i) \Pi_{t-1}^{\gamma_p}}{P_t}\right)^{-\varepsilon} \, di \\
& =(1-\theta)\left(\frac{P_t^*}{P_t}\right)^{-\varepsilon}+\Pi_{t-1}^{-\gamma_p \varepsilon} \int_{D(i)}\left(\frac{P_{t-1}}{P_t} \frac{P_{t-1}(i)}{P_{t-1}}\right)^{-\varepsilon} \, di \\
& =(1-\theta)\left(\frac{P_t^*}{P_t}\right)^{-\varepsilon}+\Pi_{t-1}^{-\gamma_p \varepsilon} \Pi_t^{\varepsilon} \int_{D(i)}\left(\frac{P_{t-1}(i)}{P_{t-1}}\right)^{-\varepsilon} \, di \\
& =(1-\theta)\left(\frac{P_t^*}{P_t}\right)^{-\varepsilon}+\theta \Pi_{t-1}^{-\gamma_p \varepsilon} \Pi_t^{\varepsilon} D_{t-1}
\end{aligned}