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Here's my guess. Let use the notation $$\tilde x_t \approx \ln(x_t) - \ln(x) \approx \dfrac{x_t - x}{x}.$$ If we take logs on both sides we get: $$\ln(G_t) = \frac{1}{1 -\rho} \ln(p_t) + \ln(y_t) - \ln(1 + p_t^{\frac{\rho}{\rho-1}})$$ Subtracting the steady state gives:  \tilde G_t = \frac{1}{1 - \rho} \tilde p_t + \tilde y_t - \left[\ln(1 + p^{\frac{\...