<p>The hint is to use the recursive Euler equation to express all future consumptions $c_{t+j}$ as a function of current consumption $c_t$, and to play with the indexes in the sum of the left.  </p>

CC BY-SA 3.0
2016-06-29T18:00:24.507Z
This would be more appropriate as a comment than an answer.
Kitsune Cavalry
CC BY-SA 3.0
2016-06-29T17:01:20.587Z
2016-06-29T17:01:20.587Z
GuiWil
<p><strong>Hint:</strong>
Separate the consumption time periods
$$\begin{align}
\left(\frac{ c_ {t+1} }{c_t}\right)^\sigma &amp; = \beta (1+r) \\
\frac{ c_ {t+1} }{c_t} &amp; = \left[\beta (1+r) \right]^\frac{1}{\sigma} \\
c_ {t+1}  &amp; = c_t \left[\beta (1+r) \right]^\frac{1}{\sigma} \\
\end{align}$$</p>

<p>Notice that $(\frac{ c_ {t+2} }{c_{t+1}})^\sigma = \beta (1+r)$. So,</p>

<p>$$c_ {t+2}  = c_{t+1} \left[\beta (1+r) \right]^\frac{1}{\sigma} = c_{t} \left[\beta (1+r) \right]^{\frac{1}{\sigma} \cdot 2}$$</p>

<p>So </p>

<p>$$\sum _{t=0}^\infty \frac{c_t}{(1+r)^t}$$</p>

<p>can be expressed as a geometric series and simplified. Working that out for yourself should give you the desired answer.</p>

CC BY-SA 3.0
2016-06-30T02:39:08.533Z
How did you get the 3rd equation from the 2nd one?
XJ.C
CC BY-SA 3.0
2016-06-30T03:16:40.583Z
I made a mistake. I'll amend the solution.
Kitsune Cavalry
CC BY-SA 3.0
2016-06-30T23:39:46.227Z
Following your hint, I can get $\sum _{t=0}^\infty \frac{c_t}{(1+r)^t}=c_t\sum_{j=0}^\infty\frac{[\beta(1+r)]^\frac{j}{\sigma}}{(1+r)^j}$.  This is calculate the PV of consumption flow. LHS start from period 0 to $\infty$,while RHS,$\sum _{j=0}^\infty \frac{c_{t+j}}{(1+r)^j}$, from period t to $\infty$. I am confuse about this.
XJ.C
CC BY-SA 3.0
2016-07-01T00:03:18.220Z
I don't *think* there's any meaningful distinction between $j$ and $t$. Just different ways to denote time.
Kitsune Cavalry
CC BY-SA 3.0
2016-07-01T00:25:11.687Z
I mean that starting from period 0 and from t (Say t=9).
XJ.C
CC BY-SA 3.0
2016-06-29T20:34:12.700Z
2016-06-30T03:21:18.053Z
2016-06-30T03:21:18.053Z
Kitsune Cavalry
<p>I have an Euler equation $ (\frac{ c_ {t+1} }{c_t})^\sigma = \beta (1+r)$,where $c_t$ is the t period consumption,r is interest rate and $\beta$ is discount rate, and a budget constraint $\sum _{t=0}^\infty \frac{c_t}{(1+r)^t}=\sum _{t=0}^\infty \frac{y_t}{(1+r)^t}$, where $y_t$ is income.</p>

<p>My question is how to substitute Euler equation into budget constraint to get the following equation?</p>

<p>$c_t\sum_{j=0}^\infty\frac{[\beta(1+r)]^\frac{j}{\sigma}}{(1+r)^j}=\sum _{t=0}^\infty \frac{y_t}{(1+r)^t}$</p>

<consumer-theory>
How to get this budget constraint?
XJ.C
# How to get this budget constraint?

(Asked by: username_0 on 2016-06-29)

I have an Euler equation $ (\frac{ c\_ {t+1} }{c\_t})^\sigma = \beta (1+r)$,where $c\_t$ is the t period consumption,r is interest rate and $\beta$ is discount rate, and a budget constraint $\sum \_{t=0}^\infty \frac{c\_t}{(1+r)^t}=\sum \_{t=0}^\infty \frac{y\_t}{(1+r)^t}$, where $y\_t$ is income.

My question is how to substitute Euler equation into budget constraint to get the following equation?

$c\_t\sum\_{j=0}^\infty\frac{[\beta(1+r)]^\frac{j}{\sigma}}{(1+r)^j}=\sum \_{t=0}^\infty \frac{y\_t}{(1+r)^t}$

**Tags:** consumer-theory

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## 2 answers

### Answer by username_1 on 2016-06-29. Score: 1

The hint is to use the recursive Euler equation to express all future consumptions $c\_{t+j}$ as a function of current consumption $c\_t$, and to play with the indexes in the sum of the left.

> **Comments:**
> **username_2**: This would be more appropriate as a comment than an answer.  

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### Answer by username_2 on 2016-06-29. Score: 1

**Hint:**
Separate the consumption time periods
$$\begin{align}
\left(\frac{ c\_ {t+1} }{c\_t}\right)^\sigma & = \beta (1+r) \\
\frac{ c\_ {t+1} }{c\_t} & = \left[\beta (1+r) \right]^\frac{1}{\sigma} \\
c\_ {t+1} & = c\_t \left[\beta (1+r) \right]^\frac{1}{\sigma} \\
\end{align}$$

Notice that $(\frac{ c\_ {t+2} }{c\_{t+1}})^\sigma = \beta (1+r)$. So,

$$c\_ {t+2} = c\_{t+1} \left[\beta (1+r) \right]^\frac{1}{\sigma} = c\_{t} \left[\beta (1+r) \right]^{\frac{1}{\sigma} \cdot 2}$$

So

$$\sum \_{t=0}^\infty \frac{c\_t}{(1+r)^t}$$

can be expressed as a geometric series and simplified. Working that out for yourself should give you the desired answer.

> **Comments:**
> **username_0**: How did you get the 3rd equation from the 2nd one?  
> **username_2**: I made a mistake. I'll amend the solution.  
> **username_0**: Following your hint, I can get $\sum _{t=0}^\infty \frac{c_t}{(1+r)^t}=c_t\sum_{j=0}^\infty\frac{[\beta(1+r)]^\frac{j}{\sigma}}{(1+r)^j}$.  This is calculate the PV of consumption flow. LHS start from period 0 to $\infty$,while RHS,$\sum _{j=0}^\infty \frac{c_{t+j}}{(1+r)^j}$, from period t to $\infty$. I am confuse about this.  
> **username_2**: I don't *think* there's any meaningful distinction between $j$ and $t$. Just different ways to denote time.  
> **username_0**: I mean that starting from period 0 and from t (Say t=9).  

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