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I've started reading Jordi Galí's Monetary Policy, Inflation and the Business Cycle (2nd ed., 2015). In section 2.5.2, Galí considers an example with the following non-separable period utility function:

$$U(X_t, N_t) = \frac{X_t^{1 - \sigma} - 1}{1 - \sigma} - \frac{N_t^{1 + \varphi}}{1 + \varphi}$$

where $X_t$ is defined as

$$X_t = \begin{cases} \displaystyle \left[ (1 - \vartheta) C_t^{1 - \nu} + \vartheta \left( \frac{M_t}{P_t} \right)^{1 - \nu} \right]^{\frac{1}{1 - \nu}} & \nu \ne 1 \\ \displaystyle C_t^{1 - \vartheta} \left( \frac{M_t}{P_t} \right)^\vartheta & \nu = 1\end{cases}$$

Galí notes on p. 35 that the household's optimality conditions are

$$\frac{W_t}{P_t} = N_t^\varphi X_t^{\sigma-\nu} C_t^\nu (1 - \vartheta)^{-1}$$

$$Q_t = \beta E_t \left\{ \left( \frac{C_{t+1}}{C_t} \right)^{-\nu} \left( \frac{X_{t+1}}{X_t} \right)^{\nu-\sigma} \left( \frac{P_t}{P_{t+1}} \right) \right\}$$

$$\frac{M_t}{P_t} = C_t(1 - \exp\{-i_t\})^{-\frac1\nu} \left( \frac{\vartheta}{1 - \vartheta}\right)^{\frac1\nu}$$

(these are his equations (37) to (39); here $Q_t$ denotes the bond price in $t$, and bonds are assumed to pay one unit of money at maturity). Galí proceeds to log-linearize (37) and (39) as follows:

$$w_t - p_t = \sigma c_t + \varphi n_t + (\nu - \sigma) (c_t - x_t)$$

$$m_t - p_t = c_t - \eta i_t$$

(each up to an additive constant, and with $\eta = \frac1{\nu(\exp(i)-1)}$.) So far, so good.

Now, on page 36, Galí writes that "[l]og-linearizing the definition of $X_t$ around [the zero-inflation] steady state and combining the resulting expression with (39) [..] yields":

$$w_t - p_t = \sigma c_t + \varphi n_t + \chi (\nu - \sigma) (c_t - (m_t - p_t))$$

with $\chi = \frac{ \vartheta^{\frac1\nu} (1 - \beta)^{1 - \frac1\nu} }{ (1 - \vartheta)^{\frac1\nu} + \vartheta^{\frac1\nu}(1 - \beta)^{1 - \frac1\nu} }$.

I'm an undergrad student, not really familiar with log-linearization techniques yet, and I don't see how this follows, or how one would go about log-linearizing $X_t$ around the steady state in the first place. I've read Zietz's Log-Linearizing Around the Steady State: A Guide with Examples, but have failed in applying the techniques described therein (to the case of $\nu \ne 1$) to match Galí's result.

Could someone be so kind and fill in the details that Galí left out? Thank you.

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    $\begingroup$ (This should be tagged microfoundations, perhaps?) $\endgroup$ – chsk Aug 11 at 13:23

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