# Quasi-linear utility. Deriving demand

I was trying to derive a general demand for the good $$x$$ for this quasi-linear function $$u(x,y) = y + 2\sqrt{x}$$ subject to standard budget constraint $$p_x x + p_y y \leq M$$

Using Kuhn-Tucker conditions (my aim was to account the corner solutions) I obtained the following results:

$$x = 0$$, $$y = M / p_y$$ when $$MRS < p_x / p_y$$

$$x = M / p_x$$, $$y = 0$$ when $$MRS > p_x / p_y$$

$$x = (p_y / p_x) ^ 2, y = M / p_y - p_y / p_x$$ when $$MRS = p_x / p_y$$

The problem is that I do not how to write down the demand for $$x$$ with all conditions listed above ($$MRS\lesseqgtr$$ ) in an explicit form. In other words, I think it is possible to be left only with prices and income as conditions for the demand.

I hope that I was clear

• The natural first step would be to simply calculate the MRS. – Michael Greinecker Sep 7 '20 at 9:09