<p>Let <span class="math-container">$p_t = \dfrac{1}{\prod_{i = 0}^{t-1}(1+r_i)}$</span>.</p>
<p>Multiply <span class="math-container">$s_{t+1} - (1+r_t) s_t = y_t - c_t$</span> by <span class="math-container">$p_{t+1}$</span> to get:
<span class="math-container">$$
p_{t+1} s_{t+1} - p_{t} s_t = p_{t+1} y_t - p_{t+1} c_t
$$</span>
Now add over all <span class="math-container">$t = 0, 1, \ldots, T$</span> (and notice that we get a telescoping sum):
<span class="math-container">$$
p_{T+1} s_{T+1} = \sum_{t = 0}^T p_{t+1} y_t - \sum_{t = 0}^T p_{t+1} c_t.
$$</span>
This assumes that <span class="math-container">$s_0 = 0$</span>.</p>
<p>Usually, one also imposes a no-ponzi condition (<span class="math-container">$\lim_{T \to \infty} p_T s_{T} = 0$</span>). If so, (and if the limits of the sums are well defined, i.e. bounded), we can take limits for <span class="math-container">$T \to \infty$</span> of above identity to obtain:
<span class="math-container">$$
\sum_{t = 0}^\infty p_{t+1} c_t = \sum_{t = 0}^\infty p_{t+1} y_t.
$$</span>
So I guess your index on <span class="math-container">$p$</span> is one off.</p>
<p>Alternatively if the budget constraint is:
<span class="math-container">$$
s_{t+1} = (y_t - c_t + s_t)(1+r_t),
$$</span>
then you would get
<span class="math-container">$$
\sum_{t = 0}^\infty p_{t} c_t = \sum_{t = 0}^\infty p_{t} y_t.
$$</span>
The difference is that here you get your income and consume at the start of the period (before you receive interest) whereas in your specification, you first receive interest and only afterwards receive income and consume.</p>

CC BY-SA 4.0
2021-09-12T15:10:12.650Z
You are right. My definition for $p_t$ was wrong. But thanks, I get the idea now!
Arturo Sbr
CC BY-SA 4.0
2021-09-12T08:19:39.550Z
2021-09-12T08:19:39.550Z
tdm
<p>Suppose an agent receives a fixed (exogenous) income every period, <span class="math-container">$y_t$</span>, and can either spend it on <span class="math-container">$c_t$</span> or save it/borrow (<span class="math-container">$s_{t+1}$</span>). Suppose also that the agent starts off with no savings or debt.</p>
<p>How do you get from this:</p>
<p><span class="math-container">$c_t + s_{t+1} = y_t + (1 + r_t) s_t$</span></p>
<p>To this:</p>
<p><span class="math-container">$\sum_{t=0}^\infty c_t p_t = \sum_{t=0}^\infty y_t p_t$</span></p>
<p>where <span class="math-container">$p_t = \frac{1}{(1 + r_1)(1 + r_2)...(1 + r_{t})}$</span></p>

<macroeconomics><intertemporal-choice>
How is the intertemporal budget constraint derived from all intratemporal constraints
Arturo Sbr
# How is the intertemporal budget constraint derived from all intratemporal constraints

(Asked by: username_0 on 2021-09-11)

Suppose an agent receives a fixed (exogenous) income every period, $y\_t$, and can either spend it on $c\_t$ or save it/borrow ($s\_{t+1}$). Suppose also that the agent starts off with no savings or debt.

How do you get from this:

$c\_t + s\_{t+1} = y\_t + (1 + r\_t) s\_t$

To this:

$\sum\_{t=0}^\infty c\_t p\_t = \sum\_{t=0}^\infty y\_t p\_t$

where $p\_t = \frac{1}{(1 + r\_1)(1 + r\_2)...(1 + r\_{t})}$

**Tags:** macroeconomics, intertemporal-choice

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## 1 answers

### Answer by username_1 on 2021-09-12. Score: 3 | Accepted answer

Let $p\_t = \dfrac{1}{\prod\_{i = 0}^{t-1}(1+r\_i)}$.

Multiply $s\_{t+1} - (1+r\_t) s\_t = y\_t - c\_t$ by $p\_{t+1}$ to get:
$$
p\_{t+1} s\_{t+1} - p\_{t} s\_t = p\_{t+1} y\_t - p\_{t+1} c\_t
$$
Now add over all $t = 0, 1, \ldots, T$ (and notice that we get a telescoping sum):
$$
p\_{T+1} s\_{T+1} = \sum\_{t = 0}^T p\_{t+1} y\_t - \sum\_{t = 0}^T p\_{t+1} c\_t.
$$
This assumes that $s\_0 = 0$.

Usually, one also imposes a no-ponzi condition ($\lim\_{T \to \infty} p\_T s\_{T} = 0$). If so, (and if the limits of the sums are well defined, i.e. bounded), we can take limits for $T \to \infty$ of above identity to obtain:
$$
\sum\_{t = 0}^\infty p\_{t+1} c\_t = \sum\_{t = 0}^\infty p\_{t+1} y\_t.
$$
So I guess your index on $p$ is one off.

Alternatively if the budget constraint is:
$$
s\_{t+1} = (y\_t - c\_t + s\_t)(1+r\_t),
$$
then you would get
$$
\sum\_{t = 0}^\infty p\_{t} c\_t = \sum\_{t = 0}^\infty p\_{t} y\_t.
$$
The difference is that here you get your income and consume at the start of the period (before you receive interest) whereas in your specification, you first receive interest and only afterwards receive income and consume.

> **Comments:**
> **username_0**: You are right. My definition for $p_t$ was wrong. But thanks, I get the idea now!  

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