# demand function from utility

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)

• Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Sep 10, 2018 at 9:15
• It's tricky to understand the question. What do MU, M and P mean? Sep 10, 2018 at 10:05
• I think you already have the demand for $x$, when you derive $x=\frac{P^2_x}{P^2_y}$. Verify by taking your demand for $y$ and plugging it back into the budget constraint to get $x$. Sep 17, 2018 at 19:38

Note that the exogenous variables to the problem are prices $$P_x$$, $$P_y$$ and income $$M$$, and your expression of $$x$$ is already expressed only with $$P_x$$, $$P_y$$.
That is, the demand for $$x$$ will be $$x(P_x, P_y, M) = \frac{P_x^2}{P_y^2}$$ , which does not depend on income level $$M$$.
On the other hand, your demand for $$y$$ $$y(P_x, P_y, M) = \frac{M}{P_y} - \frac{P_y}{P_x}$$ , depends on all the exogenous variables.
Additionally note that, if the situation happen to be $$\frac{M}{P_y} - \frac{P_y}{P_x} <0$$, which means the slope of your budget line is too flat to reduce the consumption level $$y$$, then you have no other choice but to consume $$y=0$$ and $$x=\frac{M}{P_x}$$ (the whole income is spent on $$x$$), which is called as 'corner solution' (not 'interior solution').