<p>Rather than write a complete answer, I am going to post a couple of hints that will hopefully get you underway.</p>

<hr>

<p>For part (i), note that</p>

<ul>
<li>You have two equations: $\dfrac{MU_x}{MU_y} = \dfrac{P_x}{P_y}$ and $I = P_x \cdot x + P_y \cdot y$.</li>
<li>Since we know $MU_x$, $MU_y$, $P_x$, $P_y$, and $I$, these two equations contain two unknowns.</li>
<li>Usually, so long as we have at least as many equations as there are unknown variables, we can find the value of those variables by simultaneously solving the equations.</li>
<li>the $x$ in $I = P_x \cdot x + P_y \cdot y$ is the quantity of hot dogs that you are asked to compute in part (i), so ultimately you are going to have to solve for the value of this variable.</li>
<li>my suggestion would therefore be that you try substituting into your two equations all of the things whose value you know and then try solving for $x$.</li>
</ul>

<hr>

<p>Since a budget line is a straight line, it is characterised by an intercept, $c$, and a slope, $m$.</p>

<p>If you think about drawing this line geometrically then you should recognise that</p>

<ul>
<li>the intercept is the amount of $y$ you could afford if you didn't buy any $x$ and spent all of your available money on $y$.</li>
<li>the slope is the number of units of $y$ you would have to give up to save enough money to afford one more unit of $x$.</li>
</ul>

<p>Once you know the slope and the intercept, you can write down the equation for your budget line, $y=mx+c$, and answering part (ii) is then just a matter of algebra.</p>

CC BY-SA 3.0
2016-08-27T08:27:01.540Z
Thank you. For the formula $\dfrac{MU_x}{MU_y} = \dfrac{P_x}{P_y}$ I'm unsure how to get $MU_x$ and $MU_y$. When using $0.5 \sqrt{\dfrac{Y}{X}}$ is X and Y \$1.00 and \$1.25 respectively? OR is it how many of X and Y Matthew can afford with his budget of \$20. Namely, is $X = 20$ and $Y=16$? Thanks again. Once I know this, it's just easy substitution I think (as you said).
Rubicon
CC BY-SA 3.0
2016-08-27T11:00:23.290Z
@GeekbabeTV You don't need to do either. Just leave the values of $x$ and $y$ undefined and, once you substitute the expression for marginal utility into $MU_x/MU_y$, as well as substituting in the values for prices and income you will find that you have two equations that depend only on $x$ and $y$. Then you can solve for those two variables. Note that there appears to be a typo: $x$ and $X$ are actually the same variable.
Ubiquitous
CC BY-SA 3.0
2016-08-27T06:59:24.213Z
2016-08-27T06:59:24.213Z
Ubiquitous
<p>Matthew has a utility function $U = \sqrt{XY}$ where X represents hot dogs and Y represents apples. He has an income of $20. </p>

<p>Hot dogs cost \$1.00 and apples cost \$1.25</p>

<p><strong>Part (i) asks:</strong> Matthew's utility fucntion implies that the marginal utility of apples is $0.5 \sqrt{\frac{X}{Y}}$ and the marginal utility of hotdogs is $0.5 \sqrt{\frac{Y}{X}}$. How many apples will he buy? How many hot dogs will he buy?</p>

<p><strong>Part (ii) asks:</strong> This year, the price of hot dogs rise to \$3 each while Mathew’s income is unchanged. Mathew’s father decides to help him by giving him a gift of \$20. Consider an indifference curve-budget line diagram with hot dogs on the x-axis and apples on the y-axis. After Mathew gets the \$20 gift, will his new budget line lie above, lie below, or pass through his initial optimum? Justify your choice. Will Mathew be better or worse off than he was last year?</p>

<h2>What have I done?</h2>

<p>I'm pretty stuck on this. There aren't really any similar examples from me to draw upon from our course literature. </p>

<p>Perhaps the formulas $\dfrac{MU_x}{MU_y} = \dfrac{P_x}{P_y}$ and $I = P_x \cdot x + P_y \cdot y$ might help.</p>

<microeconomics><utility>
Utility function is given as $U = \sqrt{XY}$ find
Rubicon
# Utility function is given as $U = \sqrt{XY}$ find

(Asked by: username_0 on 2016-08-27)

Matthew has a utility function $U = \sqrt{XY}$ where X represents hot dogs and Y represents apples. He has an income of $20.

Hot dogs cost \$1.00 and apples cost \$1.25

**Part (i) asks:** Matthew's utility fucntion implies that the marginal utility of apples is $0.5 \sqrt{\frac{X}{Y}}$ and the marginal utility of hotdogs is $0.5 \sqrt{\frac{Y}{X}}$. How many apples will he buy? How many hot dogs will he buy?

**Part (ii) asks:** This year, the price of hot dogs rise to \$3 each while Mathew’s income is unchanged. Mathew’s father decides to help him by giving him a gift of \$20. Consider an indifference curve-budget line diagram with hot dogs on the x-axis and apples on the y-axis. After Mathew gets the \$20 gift, will his new budget line lie above, lie below, or pass through his initial optimum? Justify your choice. Will Mathew be better or worse off than he was last year?

## What have I done?

I'm pretty stuck on this. There aren't really any similar examples from me to draw upon from our course literature.

Perhaps the formulas $\dfrac{MU\_x}{MU\_y} = \dfrac{P\_x}{P\_y}$ and $I = P\_x \cdot x + P\_y \cdot y$ might help.

**Tags:** microeconomics, utility

---

## 1 answers

### Answer by username_1 on 2016-08-27. Score: 0 | Accepted answer

Rather than write a complete answer, I am going to post a couple of hints that will hopefully get you underway.

---

For part (i), note that

* You have two equations: $\dfrac{MU\_x}{MU\_y} = \dfrac{P\_x}{P\_y}$ and $I = P\_x \cdot x + P\_y \cdot y$.
* Since we know $MU\_x$, $MU\_y$, $P\_x$, $P\_y$, and $I$, these two equations contain two unknowns.
* Usually, so long as we have at least as many equations as there are unknown variables, we can find the value of those variables by simultaneously solving the equations.
* the $x$ in $I = P\_x \cdot x + P\_y \cdot y$ is the quantity of hot dogs that you are asked to compute in part (i), so ultimately you are going to have to solve for the value of this variable.
* my suggestion would therefore be that you try substituting into your two equations all of the things whose value you know and then try solving for $x$.

---

Since a budget line is a straight line, it is characterised by an intercept, $c$, and a slope, $m$.

If you think about drawing this line geometrically then you should recognise that

* the intercept is the amount of $y$ you could afford if you didn't buy any $x$ and spent all of your available money on $y$.
* the slope is the number of units of $y$ you would have to give up to save enough money to afford one more unit of $x$.

Once you know the slope and the intercept, you can write down the equation for your budget line, $y=mx+c$, and answering part (ii) is then just a matter of algebra.

> **Comments:**
> **username_0**: Thank you. For the formula $\dfrac{MU_x}{MU_y} = \dfrac{P_x}{P_y}$ I'm unsure how to get $MU_x$ and $MU_y$. When using $0.5 \sqrt{\dfrac{Y}{X}}$ is X and Y \$1.00 and \$1.25 respectively? OR is it how many of X and Y Matthew can afford with his budget of \$20. Namely, is $X = 20$ and $Y=16$? Thanks again. Once I know this, it's just easy substitution I think (as you said).  
> **username_1**: @GeekbabeTV You don't need to do either. Just leave the values of $x$ and $y$ undefined and, once you substitute the expression for marginal utility into $MU_x/MU_y$, as well as substituting in the values for prices and income you will find that you have two equations that depend only on $x$ and $y$. Then you can solve for those two variables. Note that there appears to be a typo: $x$ and $X$ are actually the same variable.  

---

