<p>I actually tried the following:
<span class="math-container">$$
c_{0}+k_{1}=Rk_{0}\implies k_{1}=Rk_{0}-c_{0}
$$</span>
<span class="math-container">$$
k_{2}=R^{2}k_{0}-Rc_{0}-c_{1}
$$</span>
and so on. This gives us:
<span class="math-container">$$
c_{T}+R^{T}c_{0}+R^{T-1}c_{1}+...+Rc_{T-1}=R^{T+1}k_{0}
$$</span>
Multiply both sides by <span class="math-container">$\frac{1}{R^{T}}$</span>
<span class="math-container">$$
c_{0}+\frac{c_{1}}{R}+....+\frac{c_{T}}{R^{T}}=Rk_{0}
$$</span></p>

CC BY-SA 4.0
2022-08-29T15:55:07.583Z
2022-08-29T15:55:07.583Z
Kwame Brown
<p>Suppose I have the period budget constraints, where <span class="math-container">$c$</span> is consumption
and <span class="math-container">$k$</span> is capital:
<span class="math-container">$$
c_{0}+k_{1}=Rk_{0}
$$</span>
<span class="math-container">$$
c_{1}+k_{2}=Rk_{1}
$$</span>
<span class="math-container">$$
\vdots
$$</span>
<span class="math-container">$$
c_{T}+k_{T+1}=Rk_{T}
$$</span>
and
<span class="math-container">$$
k_{T+1}=0
$$</span>
The notes say that with repeated substitution, the consolidated intertemporal
budget constraint should be:
<span class="math-container">$$
c_{1}+\frac{1}{R}c_{1}+\frac{1}{R^{2}}c_{2}+...+\frac{1}{R^{T}}c_{T}=Rk_{0}
$$</span>
I am at a loss at how to get this. I tried substituting:
<span class="math-container">$$
k_{1}=Rk_{0}-c_{0}
$$</span>
and
<span class="math-container">$$
k_{2}=Rk_{1}-c_{1}=R^{2}k_{0}-Rc_{1}-c_{1}
$$</span>
and so on, but this just seems to get messier. I think the terminal
condition, i.e.
<span class="math-container">$$
c_{T}=Rk_{T}
$$</span>
is probably useful here, but I am not sure how to work backwards
with it. Any help on this is much appreciated!</p>

CC BY-SA 4.0
2022-08-29T14:48:24.233Z
Try $k_1=\frac{c_1}{R}+\frac{k_2}{R}$
Herr K.
<macroeconomics>
Deriving intertemporal budget constraint
Kwame Brown
# Deriving intertemporal budget constraint

(Asked by: username_0 on 2022-08-29)

Suppose I have the period budget constraints, where $c$ is consumption
and $k$ is capital:
$$
c\_{0}+k\_{1}=Rk\_{0}
$$
$$
c\_{1}+k\_{2}=Rk\_{1}
$$
$$
\vdots
$$
$$
c\_{T}+k\_{T+1}=Rk\_{T}
$$
and
$$
k\_{T+1}=0
$$
The notes say that with repeated substitution, the consolidated intertemporal
budget constraint should be:
$$
c\_{1}+\frac{1}{R}c\_{1}+\frac{1}{R^{2}}c\_{2}+...+\frac{1}{R^{T}}c\_{T}=Rk\_{0}
$$
I am at a loss at how to get this. I tried substituting:
$$
k\_{1}=Rk\_{0}-c\_{0}
$$
and
$$
k\_{2}=Rk\_{1}-c\_{1}=R^{2}k\_{0}-Rc\_{1}-c\_{1}
$$
and so on, but this just seems to get messier. I think the terminal
condition, i.e.
$$
c\_{T}=Rk\_{T}
$$
is probably useful here, but I am not sure how to work backwards
with it. Any help on this is much appreciated!

**Tags:** macroeconomics
> **Comments:**
> **username_1**: Try $k_1=\frac{c_1}{R}+\frac{k_2}{R}$  

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## 1 answers

### Answer by username_0 on 2022-08-29. Score: 1

I actually tried the following:
$$
c\_{0}+k\_{1}=Rk\_{0}\implies k\_{1}=Rk\_{0}-c\_{0}
$$
$$
k\_{2}=R^{2}k\_{0}-Rc\_{0}-c\_{1}
$$
and so on. This gives us:
$$
c\_{T}+R^{T}c\_{0}+R^{T-1}c\_{1}+...+Rc\_{T-1}=R^{T+1}k\_{0}
$$
Multiply both sides by $\frac{1}{R^{T}}$
$$
c\_{0}+\frac{c\_{1}}{R}+....+\frac{c\_{T}}{R^{T}}=Rk\_{0}
$$


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