<p>In your case there could be multiple solutions to the problem not just <span class="math-container">$y=0$</span>. Any combination of <span class="math-container">$x$</span> and <span class="math-container">$y$</span> that would satisfy the budget with equality would be optimal in this case.</p>
<p>Hence I think it would probably be best just express demand for <span class="math-container">$x$</span> as:</p>
<p><span class="math-container">$x=\frac{I}{P_x} - 4 y$</span></p>
<p>In fact this also gives you all potential optimum solutions to the problem since you can verify that for some parameters for example <span class="math-container">$I=100$</span> and <span class="math-container">$P_x=1$</span>, when you consume 0 units of <span class="math-container">$y$</span> (implying that <span class="math-container">$x=100$</span>) you get exactly the same utility as when you consume <span class="math-container">$x=96$</span> and <span class="math-container">$y=1$</span>, or <span class="math-container">$x=92$</span> and <span class="math-container">$y=2$</span> and so on. So the <span class="math-container">$x= \frac{I}{P_x}-4y$</span> gives the combination of all possible optimum solutions. I personally don't see an reason to just assume that <span class="math-container">$y$</span> should be zero in such situation (unless this is for a classroom setting and that was recommended by instructor in these situations).</p>

CC BY-SA 4.0
2020-11-03T13:34:56.123Z
2020-11-03T13:34:56.123Z
1muflon1
<p><span class="math-container">$U(x,y) = x + 4y$</span>, I tried to find demand function for good x, so I did utility maximization.</p>
<p>max <span class="math-container">$U(x,y)  = x + 4y $</span> subjects to <span class="math-container">$P_x.x + P_y . y = I$</span></p>
<p>and I found the <span class="math-container">$Px /Py  = 1/4$</span>, so <span class="math-container">$4Px = Py$</span>.</p>
<p>Plug it in budget constraint I found <span class="math-container">$P_x.x+4P_x.y = I$</span></p>
<p>My Question is when I write demand function for good x, should it be <span class="math-container">$x = I/P_x$</span> because I said <span class="math-container">$y = 0$</span></p>
<p>So, Can I say that there is corner solution?</p>

<microeconomics>
Demand function for good x
Jumbo09
# Demand function for good x

(Asked by: username_0 on 2020-11-03)

$U(x,y) = x + 4y$, I tried to find demand function for good x, so I did utility maximization.

max $U(x,y) = x + 4y $ subjects to $P\_x.x + P\_y . y = I$

and I found the $Px /Py = 1/4$, so $4Px = Py$.

Plug it in budget constraint I found $P\_x.x+4P\_x.y = I$

My Question is when I write demand function for good x, should it be $x = I/P\_x$ because I said $y = 0$

So, Can I say that there is corner solution?

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2020-11-03. Score: 1 | Accepted answer

In your case there could be multiple solutions to the problem not just $y=0$. Any combination of $x$ and $y$ that would satisfy the budget with equality would be optimal in this case.

Hence I think it would probably be best just express demand for $x$ as:

$x=\frac{I}{P\_x} - 4 y$

In fact this also gives you all potential optimum solutions to the problem since you can verify that for some parameters for example $I=100$ and $P\_x=1$, when you consume 0 units of $y$ (implying that $x=100$) you get exactly the same utility as when you consume $x=96$ and $y=1$, or $x=92$ and $y=2$ and so on. So the $x= \frac{I}{P\_x}-4y$ gives the combination of all possible optimum solutions. I personally don't see an reason to just assume that $y$ should be zero in such situation (unless this is for a classroom setting and that was recommended by instructor in these situations).


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