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$U(x,y) = x + 4y$, 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?
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).
