# Quasi-linear Optimal Consumption Bundle

I have a question involving optimal consumption bundles for quasi-linear preferences. Utility is given by

$$U(x_1,x_2) = 16\sqrt{x_1} + 2x_2$$

and $$p_1 = 8, p_2 = 4, I = 30$$.

What I have so far is

$$8x_1 + 4x_2 = 30$$ and

$$MRS = -\frac{4}{\sqrt{x_1}} = -\frac{p_1}{p_2} = -2$$

Solving that I get $$x_1 = 4$$, but plugging that into the budget equation results in a negative value for $$x_2$$. What am I doing wrong here?

• Quasilinear utilities can admit corner solutions also. You need to satisfy all the first order Kuhn Tucker conditions. – superhulk Oct 9 '18 at 3:33