# Log-linearizing a non-separable utility function around the steady state

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.

• (This should be tagged microfoundations, perhaps?) – chsk Aug 11 '19 at 13:23