<p>I will only provide a general outline here since this is clearly a homework question. </p>

<hr>

<p>You know that you should have, at an optimal point, that the marginal utility of each of your two goods is equal and so you can use: </p>

<p>$$MU_x=MU_y$$</p>

<p>You also have a budget constraint, yes? something like: </p>

<p>$$x+y=P_x(initial endowment of x) + P_y(initial endowment of y) + Income$$</p>

<hr>

<p>Now you have enough equations to solve, I think. Isolate either x or y using your budget constraint. To be concrete here, I will assume you isolate x. </p>

<p>Take this representation of x and plug it in for x where you've equated your marginal utilities. This should allow you to solve for y in terms of exogenous factors. Take this representation of y and plug it back into $MU_x$. This gives both x and y in terms of exogenous factors. </p>

<p>So, all you are doing is equating marginal utilities and also using the budget constraint to express the optimal bundle (optimal amounts of goods x,y) in terms of exogenous factors. </p>

CC BY-SA 3.0
2016-12-13T15:39:54.887Z
2016-12-13T15:39:54.887Z
123
<p>U=(xy)
du/dx=y
du/dy=x
Mux=y
Muy=x
Then Budget constraints =Px.X+Py.Y=M
Price ratios px/py=y/x
1000/500=y/x
Cross multiply when you get y, put it in the budget constraints and solve for y and use it to solve for x
When you finish, equate x and y values to the price ratio to get the optimal consumption bundle. </p>

CC BY-SA 3.0
2017-10-20T19:07:44.407Z
2017-10-20T19:07:44.407Z
Richmond
<p>The utility function of the consumer is $5x^{0.5}y^{0.5}$  </p>

<p>The income of the consumer is $5000$</p>

<p>The price of good $x$ is $1,000$ and the price of good $y$ is $500$.</p>

<p>Determine the optimal consumption bundle. </p>

<hr>

<p>Using the formula $\frac{MU_x}{P_x}=\frac{MU_y}{P_y}$ (1) I have derived the functions:</p>

<p>$MU_x = 2.5x^{-0.5}y^{0.5}$</p>

<p>$MU_y= 2.5x^{0.5}y^{-0.5}$</p>

<p>Substituting them into (1) gives:</p>

<p>$\frac{2.5x^{-0.5}y^{0.5}}{1000} = \frac{2.5x^{0.5}y^-{0.5}}{500}$</p>

<p>I'm not sure where to go from here?</p>

<p>Thank you in advance</p>

CC BY-SA 3.0
2016-12-12T15:16:47.320Z
You have one equation in two unknown! In order to find the solution you need one more equation, the budget constraint!
PostDocing
<microeconomics><consumer-theory><self-study>
Determining optimal consumption bundle
ROD
# Determining optimal consumption bundle

(Asked by: username_0 on 2016-12-12)

The utility function of the consumer is $5x^{0.5}y^{0.5}$

The income of the consumer is $5000$

The price of good $x$ is $1,000$ and the price of good $y$ is $500$.

Determine the optimal consumption bundle.

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Using the formula $\frac{MU\_x}{P\_x}=\frac{MU\_y}{P\_y}$ (1) I have derived the functions:

$MU\_x = 2.5x^{-0.5}y^{0.5}$

$MU\_y= 2.5x^{0.5}y^{-0.5}$

Substituting them into (1) gives:

$\frac{2.5x^{-0.5}y^{0.5}}{1000} = \frac{2.5x^{0.5}y^-{0.5}}{500}$

I'm not sure where to go from here?

Thank you in advance

**Tags:** microeconomics, consumer-theory, self-study
> **Comments:**
> **username_1**: You have one equation in two unknown! In order to find the solution you need one more equation, the budget constraint!  

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## 2 answers

### Answer by username_2 on 2016-12-13. Score: 1

I will only provide a general outline here since this is clearly a homework question.

---

You know that you should have, at an optimal point, that the marginal utility of each of your two goods is equal and so you can use:

$$MU\_x=MU\_y$$

You also have a budget constraint, yes? something like:

$$x+y=P\_x(initial endowment of x) + P\_y(initial endowment of y) + Income$$

---

Now you have enough equations to solve, I think. Isolate either x or y using your budget constraint. To be concrete here, I will assume you isolate x.

Take this representation of x and plug it in for x where you've equated your marginal utilities. This should allow you to solve for y in terms of exogenous factors. Take this representation of y and plug it back into $MU\_x$. This gives both x and y in terms of exogenous factors.

So, all you are doing is equating marginal utilities and also using the budget constraint to express the optimal bundle (optimal amounts of goods x,y) in terms of exogenous factors.


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### Answer by username_3 on 2017-10-20. Score: -1

U=(xy)
du/dx=y
du/dy=x
Mux=y
Muy=x
Then Budget constraints =Px.X+Py.Y=M
Price ratios px/py=y/x
1000/500=y/x
Cross multiply when you get y, put it in the budget constraints and solve for y and use it to solve for x
When you finish, equate x and y values to the price ratio to get the optimal consumption bundle.


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