<p>We have the utility function.</p>

<p><span class="math-container">$$U_{t} = \ln{c_{t}} + E_{t}\sum_{s=1}^{\infty}(\beta^{s}\ln{c_{t+s}})$$</span>
And I am trying to find the value function.</p>

<p><span class="math-container">$U$</span> is utility function. <span class="math-container">$c_t$</span> is consumption at time <span class="math-container">$t$</span>. <span class="math-container">$\beta$</span> is the discount factor.</p>

<p>My manual says multiply the right side with <span class="math-container">$1-\beta$</span> to get:</p>

<p><span class="math-container">$$U_{t} = (1-\beta)\ln c_{t} + \beta E_{t}U_{t+1}$$</span></p>

<p>How? And multiply only the right side? I am unsure what is happening here and how.</p>

<p>For <span class="math-container">$s = 1$</span> we have</p>

<p><span class="math-container">$$U_{t} = \ln c_{t} + \beta E_{t}\ln c_{t+1}$$</span></p>

<p>but this is as far as I can understand. I don't get anywhere multiplying the equation by <span class="math-container">$(1-\beta)$</span>.</p>

CC BY-SA 4.0
2019-10-15T21:15:42.927Z
I believe multiplying only the right side would be considered a *positive monotonic transformation*. So it would preserve the underlying preferences. This implies that the indifference curves are the same and the ordering of the 'bundles' are the same as before. I am not sure but I imagine this would imply that the value functions would be identical? That would be a good place to look into while waiting for someone to help you with this!
Brennan
CC BY-SA 4.0
2019-10-16T02:24:34.067Z
Hi! Just to let you know you could use $\LaTeX$ to format equations in SE.
Art
CC BY-SA 4.0
2019-10-16T02:32:27.520Z
@Art I provided an edit doing just that. Just awaiting OP approval
Brennan
CC BY-SA 4.0
2019-10-16T14:47:03.930Z
I tried latex but nothing happened in the preview
MyJAJAJAJJA
CC BY-SA 4.0
2019-10-16T14:47:27.247Z
But even if multiplying only one side, I can't understand how we end up here
MyJAJAJAJJA
<mathematical-economics><utility><optimization><expected-utility>
derive value function from utility function
MyJAJAJAJJA
# derive value function from utility function

(Asked by: username_0 on 2019-10-15)

We have the utility function.

$$U\_{t} = \ln{c\_{t}} + E\_{t}\sum\_{s=1}^{\infty}(\beta^{s}\ln{c\_{t+s}})$$
And I am trying to find the value function.

$U$ is utility function. $c\_t$ is consumption at time $t$. $\beta$ is the discount factor.

My manual says multiply the right side with $1-\beta$ to get:

$$U\_{t} = (1-\beta)\ln c\_{t} + \beta E\_{t}U\_{t+1}$$

How? And multiply only the right side? I am unsure what is happening here and how.

For $s = 1$ we have

$$U\_{t} = \ln c\_{t} + \beta E\_{t}\ln c\_{t+1}$$

but this is as far as I can understand. I don't get anywhere multiplying the equation by $(1-\beta)$.

**Tags:** mathematical-economics, utility, optimization, expected-utility
> **Comments:**
> **username_1**: I believe multiplying only the right side would be considered a *positive monotonic transformation*. So it would preserve the underlying preferences. This implies that the indifference curves are the same and the ordering of the 'bundles' are the same as before. I am not sure but I imagine this would imply that the value functions would be identical? That would be a good place to look into while waiting for someone to help you with this!  
> **username_2**: Hi! Just to let you know you could use $\LaTeX$ to format equations in SE.  
> **username_1**: @Art I provided an edit doing just that. Just awaiting OP approval  
> **username_0**: I tried latex but nothing happened in the preview  
> **username_0**: But even if multiplying only one side, I can't understand how we end up here  

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## 0 answers

