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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?

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    $\begingroup$ Quasilinear utilities can admit corner solutions also. You need to satisfy all the first order Kuhn Tucker conditions. $\endgroup$
    – superhulk
    Oct 9, 2018 at 3:33

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