# Log-linearization of an intertemporal budget constraint

I'm trying to recreate the paper "Learning about Monetary Policy Rules when Long-Horizon Expectations Matter" by Preston.

At one point he says that he log linearizes the intertemporal budget constraint

$$W_{t}^{i}=\sum_{j=0}^{\infty} R_{t, t+j}\left[P_{t+j} C_{t+j}^{i}-P_{t+j} Y_{t+j}^{i}\right]$$

where $$W_{t}^{i} \equiv\left(1+i_{t-1}\right) B_{t-1}$$, $$\bar{\imath}_{t}=\beta^{-1}-1$$ (denoting the steady state of $$i$$), $$\hat{i}=\log [(1+i) /(1+\bar{i})]= log \frac{R_{t,t+1}}{R}$$ in order to obtain $$\hat{E}_{t}^{i} \sum_{T=t}^{\infty} \beta^{T-t} \hat{C}_{T}^{i}=\varpi_{t}^{i}+\hat{E}_{t}^{i} \sum_{T=t}^{\infty} \beta^{T-t} \hat{Y}_{T}^{i}$$

with $$\varpi_{t}^{i} \equiv W_{t}^{i} /\left(P_{t} \bar{Y}\right)$$.

I tried a houndred times to linearize the budget constraint but i can't really figure out what he did.

Since $$ln x_t = ln x - \hat x_t \rightarrow x_t = x e^{\hat x_t}$$ i wrote the the intertemporal budget constraint as

\begin{align} W_t^i =&\sum_j^{\infty} R e^{\hat R_{t,t+1}} P e^{\hat P_{t+j}} \left[ ( C e^{\hat C_t+j}) - (Y e^{\hat Y_t})\right]\\ W_t^i=& \sum_j^{\infty} RP e^{\hat R_{t,t+1} + \hat P_{t+j}} \left[ ( C e^{\hat C_t+j}) - (Y e^{\hat Y_t})\right] \\ W_t^i \approx & \sum_j^{\infty} RP (1+ \hat R_{t,t+1}+ \hat P_{t+j}) \left[ (C(1+\hat C_{t+j})(Y(1+ \hat Y_{t+j})\right] \end{align} ANd then i don't know how to procede. Where do the steady state values of $$R$$ and $$C$$ disappear?

• You should show us at least one of the attempts at log linearization you did per rules of this forum to show us some effort. – 1muflon1 Jul 18 at 10:52
• I added one but i think it's completely wrong – bdvse Jul 18 at 12:08
• that’s completely fine at least you show us you tried – 1muflon1 Jul 18 at 12:09
• Haven't looked at the detailed calculations yet. But have you replaced the steady state values and conditions in your log linearized equation? If you havent done it, that could be a potential source of error. – Tomcat Jul 18 at 15:57
• Which steady state value do you suggest? My problem is that W_t^i (LHS on the first equation) also appears in the log linearized one in the term ϖ_t^i. @Tomcat – bdvse Jul 19 at 13:21