\(\pi_{ADJ} = \frac{M}{P}(\frac{\eta}{1-\eta}(\frac{M}{P})^{\gamma - 1})^{1-\eta} - (\frac{M}{P})^\gamma (\frac{\eta}{1-\eta}(\frac{M}{P})^{\gamma - 1})^{-\eta}\)
\(\pi_{ADJ} = (\frac{\eta}{1-\eta})^{1-\eta}(\frac{M}{P})^{\gamma - \gamma\eta+\eta} -(\frac{\eta}{1-\eta})^{-\eta}(\frac{M}{P})^{\gamma - \gamma\eta + \eta}\)
\(\pi_{ADJ} = (\frac{\eta}{1-\eta} -1)(\frac{\eta}{1-\eta})^{-\eta}(\frac{M}{P})^{\gamma - \gamma\eta + \eta}\)
\(\pi_{ADJ} = \frac{1}{\eta-1}(\frac{\eta}{1-\eta})^{-\eta}(\frac{M}{P})^{\gamma - \gamma\eta + \eta}\)
At the equilibrium where \(P_i = P\), we have \(Y=[\frac{1-\eta}{\eta}]^{\frac{1}{\gamma-1}}\)
\(\pi_{ADJ} = \frac{M}{P}(\frac{\eta}{1-\eta}(\frac{M}{[\frac{1-\eta}{\eta}]^{\frac{1}{\gamma-1}}})^{\gamma - 1})^{1-\eta} - (\frac{M}{P})^\gamma (\frac{\eta}{1-\eta}(\frac{M}{[\frac{1-\eta}{\eta}]^{\frac{1}{\gamma-1}}})^{\gamma - 1})^{-\eta} = \frac{M}{P} - (\frac{M}{P})^\gamma\)
In other words, when \(\frac{M}{P}\) equals its flexible price equilibrium, then \(\pi_{ADJ} = \pi_{FIXED}\).
Let's have \(\mathcal{z}\) being equal to a menu cost. Then rigidity is an equilibrium if:
\(\mathcal{z} < G_N\)
\(G_N = \pi_{ADJ} - \pi_{FIXED}\)
Ball and Romer (1993)
It had been a peaceful night on Earth, where all the women are strong, all the men are good looking, and all the children are above average. At this same date, the 24 October 2529, on Mars, the economy was far from peaceful. On the red planet, where all the women, men and children are average, a monetary shock led to a serious economic crisis.
\(\mathcal{U}_i = C_i - \frac{\eta -1}{\gamma\eta}L_i^\gamma - \mathcal{z}D_i\)
\(Y_i=L_i\)