<p><strong>Hint:</strong></p>

<p>Rearrange your equation 1. to express optimal consumption <span class="math-container">$x$</span> as a function of <span class="math-container">$P_x$</span> and <span class="math-container">$P_y$</span>!</p>

<p>Does the rhs now contain <span class="math-container">$I$</span>?</p>

CC BY-SA 4.0
2020-02-11T14:39:45.600Z
Not sure why I did not see this when I posted the question... perhaps it was a long day! But thank you :)
Ethan Mark
CC BY-SA 4.0
2020-02-10T10:57:27.910Z
2020-02-10T10:57:27.910Z
VARulle
<p><strong>hint</strong>:</p>

<p>Solve for optimal <span class="math-container">$x^*,y^*$</span> then substitute the solutions into budget constraint and then solve the budget constraint for <span class="math-container">$x^*,y^*$</span> respectively and you will get the demands as a function of the parameters and of course conditional on the optimal consumption of the other variable.</p>

<p>Knowing that we have an interior solution helps because that means we know demand(s) is(are) not zero.</p>

CC BY-SA 4.0
2020-02-09T15:25:50.157Z
But how do I solve for the optimal basket if I do not know the consumer's budget constraint? Do I just use the general equation for a budget line?
Ethan Mark
CC BY-SA 4.0
2020-02-09T15:27:24.657Z
@EthanMark yes that’s the budget constraint - you can call it budget line if it’s linear like in your case but normally I would call it budget/budget constraint (it’s the constraint that you use in your constrained optimization (Lagrangian) problem
1muflon1
CC BY-SA 4.0
2020-02-09T16:12:51.937Z
I see. Thank you! I found the demand of $x$ to be $x = \frac {(P_y)^2} {4(P_x)^2}$. Is this correct?
Ethan Mark
CC BY-SA 4.0
2020-02-09T17:22:00.107Z
@EthanMark no the correct answer will be function of $I$ common you are almost there. If you want demand for x just play optimum y into budget and solve for x, this is as direct hint as it goes only if thing left is actually solving it
1muflon1
CC BY-SA 4.0
2020-02-09T17:31:53.270Z
But the question is asking me to show that the demand of $x$ does not depend on income. If my final answer for the demand of $x$ is a function of $I$, aren’t I contradicting the question?
Ethan Mark
CC BY-SA 4.0
2020-02-09T17:33:35.887Z
@EthanMark oh sorry I did not noticed that, well my steps are correct but income should then fall out of the equation - probably it will be on both sides you will divide it and it will fall out - but ignore the part of my previous comment but the steps are still correct
1muflon1
CC BY-SA 4.0
2020-02-09T15:23:17.597Z
2020-02-09T17:07:49.360Z
2020-02-09T17:07:49.360Z
1muflon1
<p>A consumer's utility function is <span class="math-container">$U(x,y)=\sqrt x + y$</span>. Assuming we have an interior solution, I need to show that the demand for <span class="math-container">$x$</span> does not depend on income.</p>

<p>I know that the consumer consumes where the marginal utilities per dollar of <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are equal, so I have found the following two equations:</p>

<ol>
<li><span class="math-container">$\frac 1 {2P_x\sqrt x}$</span> = <span class="math-container">$\frac 1 {P_y}$</span></li>
<li><span class="math-container">${P_x}x + {P_y}y = I$</span></li>
</ol>

<p>Then, I am stuck. I do not see how I can get rid of both <span class="math-container">$y$</span> and <span class="math-container">$I$</span> in this case? And how does knowing that we have an interior solution help?</p>

CC BY-SA 4.0
2020-02-09T15:44:26.510Z
You have 2 equations and 2 unknowns, right? What is keeping you from solving for $x$ and $y$? Seems like you can get $x$ right away from the first equation. Failing to do that, try to express $x$ and $y$ as functions of $p_x,p_y$ and $I$, but not each other.
Giskard
CC BY-SA 4.0
2020-02-09T15:52:19.417Z
Okay. I took a break and tried again. I got the demand function of $x$ to be $x = \frac {(P_y)^2} {4(P_x)^2}$. Is this correct? I literally just saw that I could obtain $x$ directly from the first equation... It must have been a long day for me.
Ethan Mark
<microeconomics><utility><demand>
Derivation of a demand equation
Ethan Mark
# Derivation of a demand equation

(Asked by: username_0 on 2020-02-09)

A consumer's utility function is $U(x,y)=\sqrt x + y$. Assuming we have an interior solution, I need to show that the demand for $x$ does not depend on income.

I know that the consumer consumes where the marginal utilities per dollar of $x$ and $y$ are equal, so I have found the following two equations:

1. $\frac 1 {2P\_x\sqrt x}$ = $\frac 1 {P\_y}$
2. ${P\_x}x + {P\_y}y = I$

Then, I am stuck. I do not see how I can get rid of both $y$ and $I$ in this case? And how does knowing that we have an interior solution help?

**Tags:** microeconomics, utility, demand
> **Comments:**
> **username_1**: You have 2 equations and 2 unknowns, right? What is keeping you from solving for $x$ and $y$? Seems like you can get $x$ right away from the first equation. Failing to do that, try to express $x$ and $y$ as functions of $p_x,p_y$ and $I$, but not each other.  
> **username_0**: Okay. I took a break and tried again. I got the demand function of $x$ to be $x = \frac {(P_y)^2} {4(P_x)^2}$. Is this correct? I literally just saw that I could obtain $x$ directly from the first equation... It must have been a long day for me.  

---

## 2 answers

### Answer by username_2 on 2020-02-10. Score: 0 | Accepted answer

**Hint:**

Rearrange your equation 1. to express optimal consumption $x$ as a function of $P\_x$ and $P\_y$!

Does the rhs now contain $I$?

> **Comments:**
> **username_0**: Not sure why I did not see this when I posted the question... perhaps it was a long day! But thank you :)  

---

### Answer by username_3 on 2020-02-09. Score: 1

**hint**:

Solve for optimal $x^\*,y^\*$ then substitute the solutions into budget constraint and then solve the budget constraint for $x^\*,y^\*$ respectively and you will get the demands as a function of the parameters and of course conditional on the optimal consumption of the other variable.

Knowing that we have an interior solution helps because that means we know demand(s) is(are) not zero.

> **Comments:**
> **username_0**: But how do I solve for the optimal basket if I do not know the consumer's budget constraint? Do I just use the general equation for a budget line?  
> **username_3**: @EthanMark yes that’s the budget constraint - you can call it budget line if it’s linear like in your case but normally I would call it budget/budget constraint (it’s the constraint that you use in your constrained optimization (Lagrangian) problem  
> **username_0**: I see. Thank you! I found the demand of $x$ to be $x = \frac {(P_y)^2} {4(P_x)^2}$. Is this correct?  
> **username_3**: @EthanMark no the correct answer will be function of $I$ common you are almost there. If you want demand for x just play optimum y into budget and solve for x, this is as direct hint as it goes only if thing left is actually solving it  
> **username_0**: But the question is asking me to show that the demand of $x$ does not depend on income. If my final answer for the demand of $x$ is a function of $I$, aren’t I contradicting the question?  
> **username_3**: @EthanMark oh sorry I did not noticed that, well my steps are correct but income should then fall out of the equation - probably it will be on both sides you will divide it and it will fall out - but ignore the part of my previous comment but the steps are still correct  

---

