thanks in advance.
I’m looking to find a way (apart from Lagrange and MUx/MUy=Px/Py) to derive demand function for x, not y. You see, whenever I try solving the problem, I get y’s Demand instead of x.
I have a function, $u(x,y)= 2\sqrt{x} + y$
whenever I differentiate and then put in the budget constraint equation, I end up getting the result of differentiation of x and the demand for y.
For example, $$\ MUx = \frac{1}{ \sqrt{x} }$$
$MUy=1$,
$$\frac{MUx}{MUy} = \frac{Px}{Py}$$
$$\frac{1}{\sqrt{x}} = \frac{Px}{Py}$$
$$\ x = \frac{Py^2}{Px^2}$$
then, substituting x in Budget constraint equation yields $$\ y = \frac{M}{Py} - \frac{Py}{Px}$$ with the impossibility to find x’s demand (or maybe I just don’t understand it)
MUx, MUy are marginal utilities of goods x and y respectively
Px, Py are prices of these goods
M is income, that comes from budget constraint equation (Px * x +Py * y=M)