<p>I am studying the following problem -</p>
<p>Investor has $1 and he has to split it between two assets. These are risk-free assets.</p>
<p>Under scenario 1 - These are normal times so he don't have to worry about anything, just the net returns.
Interest from asset 1 is A and interest from asset 2 is B</p>
<p>Total returns for the investor is
<span class="math-container">$$ \sqrt(x) A + \sqrt(1-x) B $$</span></p>
<p>So, he will choose x (amount of investment in asset 1) -
<span class="math-container">$$ \frac{\sqrt(x)}{\sqrt(1-x)} = \frac{A}{B} $$</span></p>
<p>Scenario 2: These are the hard times. His investment decision has an effect on other domains of his life, denoted by y</p>
<p>His total returns is given by -
<span class="math-container">$$ \sqrt(x) (C - y^2 + xy) + \sqrt(1-x) (D - y^2 - 3y + xy) $$</span></p>
<p>So, foc will be given by -
<span class="math-container">$$\frac{1}{2\sqrt(x)} (C - y^2 + xy) - \frac{1}{2\sqrt(1-x)} (D - y^2 - 3y + xy) + \sqrt(x) ( - 2y \frac{dy}{dx} + y + x\frac{dy}{dx}) + \sqrt(1-x) (- 3\frac{dy}{dx} - 2y \frac{dy}{dx} + x\frac{dy}{dx} + y) =0 $$</span></p>
<p>I have to compare x under scenario 1 and scenario 2</p>
<p>I know that</p>
<p><span class="math-container">$$\frac{A}{B} &lt; \frac{(C - y^2 + xy)}{(D - y^2 - 3y + xy)}$$</span></p>
<p>and</p>
<p><span class="math-container">$$ 0&gt;( - 2y \frac{dy}{dx} + y + x\frac{dy}{dx}) &gt; (- 3\frac{dy}{dx} - 2y \frac{dy}{dx} + x\frac{dy}{dx} + y) $$</span></p>
<p>How can I prove that x under scenario 2 &gt; x under scenario 1?</p>
<p>Please guide me with follow up action</p>

CC BY-SA 4.0
2024-11-19T19:12:40.430Z
So the proportions invested in either asset ($x$ and $1-x$) are put under the square root but the returns (e.g. $A$ and $B$) are not? What's $\sqrt k$ in the returns in scenario 2? Was it supposed to be $\sqrt x$?
Herr K.
CC BY-SA 4.0
2024-11-20T00:51:34.270Z
@HerrK. Thanks for pointing it out, yes it's x (it was a typo). Reason I have taken sqrt of x because had it been x, he would have chosen x=1 for A>B. He wouldn't have been concerned about relative. And sqrt of x can be interpreted as investment left after paying some fees and you yet interest A on only that. Sorry, I should have mentioned it in my question.
Elina Gilbert
<microeconomics><optimization><comparative-economics><differentiation><first-order-condition>
Comparing amount of investment under two situations
Elina Gilbert
# Comparing amount of investment under two situations

(Asked by: username_0 on 2024-11-19)

I am studying the following problem -

Investor has $1 and he has to split it between two assets. These are risk-free assets.

Under scenario 1 - These are normal times so he don't have to worry about anything, just the net returns.
Interest from asset 1 is A and interest from asset 2 is B

Total returns for the investor is
$$ \sqrt(x) A + \sqrt(1-x) B $$

So, he will choose x (amount of investment in asset 1) -
$$ \frac{\sqrt(x)}{\sqrt(1-x)} = \frac{A}{B} $$

Scenario 2: These are the hard times. His investment decision has an effect on other domains of his life, denoted by y

His total returns is given by -
$$ \sqrt(x) (C - y^2 + xy) + \sqrt(1-x) (D - y^2 - 3y + xy) $$

So, foc will be given by -
$$\frac{1}{2\sqrt(x)} (C - y^2 + xy) - \frac{1}{2\sqrt(1-x)} (D - y^2 - 3y + xy) + \sqrt(x) ( - 2y \frac{dy}{dx} + y + x\frac{dy}{dx}) + \sqrt(1-x) (- 3\frac{dy}{dx} - 2y \frac{dy}{dx} + x\frac{dy}{dx} + y) =0 $$

I have to compare x under scenario 1 and scenario 2

I know that

$$\frac{A}{B} < \frac{(C - y^2 + xy)}{(D - y^2 - 3y + xy)}$$

and

$$ 0>( - 2y \frac{dy}{dx} + y + x\frac{dy}{dx}) > (- 3\frac{dy}{dx} - 2y \frac{dy}{dx} + x\frac{dy}{dx} + y) $$

How can I prove that x under scenario 2 > x under scenario 1?

Please guide me with follow up action

**Tags:** microeconomics, optimization, comparative-economics, differentiation, first-order-condition

> **Comments:**
> **username_1**: So the proportions invested in either asset ($x$ and $1-x$) are put under the square root but the returns (e.g. $A$ and $B$) are not? What's $\sqrt k$ in the returns in scenario 2? Was it supposed to be $\sqrt x$?  
> **username_0**: @HerrK. Thanks for pointing it out, yes it's x (it was a typo). Reason I have taken sqrt of x because had it been x, he would have chosen x=1 for A>B. He wouldn't have been concerned about relative. And sqrt of x can be interpreted as investment left after paying some fees and you yet interest A on only that. Sorry, I should have mentioned it in my question.  

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## 0 answers

