RDP 2009-01: Currency Misalignments and Optimal Monetary Policy: A Re-examination Equation (B17)

\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} - 2 c_{y}) ({\bar{y}}_{t} - {\bar{y}}_{t}^{*}) ({\tilde{y}}_{t} - {\tilde{y}}_{t}^{*}) + (1 - σ) ({\bar{y}}_{t} {\tilde{y}}_{t} + {\bar{y}}_{t}^{*} {\tilde{y}}_{t}^{*}) \\ - (1 - σ) (\frac{ν}{2} {\bar{c}}_{t} + (\frac{2 - ν}{2}) {\bar{c}}_{t}^{*}) {\tilde{y}}_{t} - (1 - σ) (\frac{ν}{2} {\bar{c}}_{t}^{*} + (\frac{2 - ν}{2}) {\bar{c}}_{t}) {\tilde{y}}_{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}) \\ + 2 (1 - σ) (c_{y}^{2} - c_{y}) ({\bar{y}}_{t} - {\bar{y}}_{t}^{*}) ({\tilde{y}}_{t} - {\tilde{y}}_{t}^{*}) + (1 - σ) ({\bar{y}}_{t} {\tilde{y}}_{t} + {\bar{y}}_{t}^{*} {\tilde{y}}_{t}^{*}) \\ - (1 - σ) [((ν - 1) c_{y} - \frac{ν}{2}) ({\bar{y}}_{t} - {\bar{y}}_{t}^{*}) ({\tilde{y}}_{t} - {\tilde{y}}_{t}^{*}) + {\bar{y}}_{t}^{*} {\tilde{y}}_{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}^{*}) ({\bar{y}}_{t} - {\bar{y}}_{t}^{*}) + o (‖ a^{3} ‖) . \end{array}