RDP 2009-01: Currency Misalignments and Optimal Monetary Policy: A Re-examination Equation (B39)

\begin{array}{l} \frac{1 - σ}{2} ({\tilde{c}}_{t}^{2} + {\tilde{c}}_{t}^{* 2}) - \frac{1 + ϕ}{2} ({\tilde{n}}_{t}^{2} + {\tilde{n}}_{t}^{* 2}) + (1 - σ) ({\bar{c}}_{t} {\tilde{c}}_{t} + {\bar{c}}_{t}^{*} {\tilde{c}}_{t}^{*}) - (1 + ϕ) ({\bar{n}}_{t} {\tilde{n}}_{t} + {\bar{n}}_{t}^{*} {\tilde{n}}_{t}^{*}) \\ = (\frac{1 - σ}{2}) (2 c_{y}^{2} - 2 c_{y} + 1) ({\tilde{y}}_{t}^{2} + {\tilde{y}}_{t}^{* 2}) - (1 - σ) (2 c_{y}^{2} - 2 c_{y}) {\tilde{y}}_{t} {\tilde{y}}_{t}^{*} \\ - (\frac{1 + ϕ}{2}) ({\tilde{y}}_{t}^{2} + {\tilde{y}}_{t}^{* 2}) + (1 - σ) (2 (c_{y}^{2} - c_{y}) - (ν - 1) c_{y} + \frac{ν}{2}) ({\bar{y}}_{t} - {\bar{y}}_{t}^{*}) ({\tilde{y}}_{t} - {\tilde{y}}_{t}^{*}) \\ + \frac{(1 - σ) ν^{2} {(2 - ν)}^{2}}{4 D^{2}} Δ_{t}^{2} + \frac{(1 - σ) ν (2 - ν) (ν - 1)}{2 D^{2}} Δ_{t} (y_{t} - y_{t}^{*}) + o (‖ a^{2} ‖) . \end{array}