Demand function for good x

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

1 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).