<p>I'm trying to recreate the paper &quot;Learning about Monetary Policy Rules when
Long-Horizon Expectations Matter&quot; by Preston.</p>
<p>At one point he says that he log linearizes the intertemporal budget constraint</p>
<p><span class="math-container">$$W_{t}^{i}=\sum_{j=0}^{\infty} R_{t, t+j}\left[P_{t+j} C_{t+j}^{i}-P_{t+j} Y_{t+j}^{i}\right]$$</span></p>
<p>where <span class="math-container">$W_{t}^{i} \equiv\left(1+i_{t-1}\right) B_{t-1}$</span>, <span class="math-container">$\bar{\imath}_{t}=\beta^{-1}-1$</span> (denoting the steady state of <span class="math-container">$i$</span>), <span class="math-container">$\hat{i}=\log [(1+i) /(1+\bar{i})]= log \frac{R_{t,t+1}}{R}$</span>  in order to obtain
<span class="math-container">$$\hat{E}_{t}^{i} \sum_{T=t}^{\infty} \beta^{T-t} \hat{C}_{T}^{i}=\varpi_{t}^{i}+\hat{E}_{t}^{i} \sum_{T=t}^{\infty} \beta^{T-t} \hat{Y}_{T}^{i}$$</span></p>
<p>with <span class="math-container">$\varpi_{t}^{i} \equiv W_{t}^{i} /\left(P_{t} \bar{Y}\right)$</span>.</p>
<p>I tried a houndred times to linearize the budget constraint but i can't really figure out what he did.</p>
<p>Since <span class="math-container">$ln x_t = ln x -  \hat x_t \rightarrow x_t = x  e^{\hat x_t}$</span> i wrote the the intertemporal budget constraint as</p>
<p><span class="math-container">\begin{align} 
W_t^i =&amp;\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=&amp; \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 &amp; \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}</span>
ANd then i don't know how to procede. Where do the steady state values of <span class="math-container">$R$</span> and <span class="math-container">$C$</span> disappear?</p>

CC BY-SA 4.0
2020-07-18T10:52:27.517Z
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
CC BY-SA 4.0
2020-07-18T12:08:27.580Z
I added one but i think it's completely wrong
bdvse
CC BY-SA 4.0
2020-07-18T12:09:18.630Z
that’s completely fine at least you show us you tried
1muflon1
CC BY-SA 4.0
2020-07-19T13:21:47.167Z
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
<macroeconomics>
Log-linearization of an intertemporal budget constraint
bdvse
# Log-linearization of an intertemporal budget constraint

(Asked by: username_0 on 2020-07-18)

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?

**Tags:** macroeconomics

> **Comments:**
> **username_1**: 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.  
> **username_0**: I added one but i think it's completely wrong  
> **username_1**: that’s completely fine at least you show us you tried  
> **username_0**: 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  

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## 0 answers

