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Here's the question that I really need help with:

$$U(x,y)=(x^{0.5})+y$$

$MUx=\frac{1}{2x^{0.5}}$ and $MUy=1$, where $x$ is food, $y$ is clothing

$Px$ is the price of food, $Py$ is the price of clothing and $I$ is income.

For the purposes of this question you should assume that $I/Py \ge Py/4Px$.

My question is I don't know how to find the demand function for $y$.

I use $MUx/MUy=Px/Py$ and I can only find the demand function (or opportunity set) for $x$.

Is $I/Py>=Py/4Px$ the budget constraint? If it is, that's the weirdest form I've ever seen.

I'd really appreciate it if any of you could give me a hint how to solve for this question.

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closed as off-topic by Giskard, jmbejara Aug 22 '16 at 20:08

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • $\begingroup$ I would take the budget constraint to be a natural $xP_x+yP_y \le I$ $\endgroup$ – Henry Aug 17 '16 at 13:24
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First, your utility function is strictly increasing in both goods, so you know that your budget will bind so you can set $I=p_xx+p_yx$.

Second, your utility function is known as "quasilinear", where the good y is linear in your utility and the other good is there in some increasing function not connected to y. That's why the marginal utility for y is just a constant.

Finally, solve for x as you did, where x is a function of prices, then plug that into the budget constraint to solve for y. y should be a function of prices and income.

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