Consider the utility function $u(x)=x+y^2+2z^2$. How to derive Marshallian demand for a consumer with these preferences?

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    $\begingroup$ Just construct the Lagrangean using the budget constraint $$M = p_{x}x + p_{y}y + p_{z}z$$ and take the first order conditions and solve each. The result will be the Marshallian demand. $\endgroup$ – Brennan Dec 13 '19 at 2:04
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    $\begingroup$ @Brennan Does the convexity of this function not matter at all, are you sure first order conditions are sufficient? $\endgroup$ – Giskard Dec 13 '19 at 5:36
  • $\begingroup$ That's why I'm asking, I can't get far with the usual interior conditions. $\endgroup$ – econ86 Dec 13 '19 at 9:02
  • $\begingroup$ @Giskard very good point that I overlooked $\endgroup$ – Brennan Dec 13 '19 at 17:52

Just compare the utilities of these three bundles: $\left(\frac{M}{p_X}, 0, 0\right)$, $\left(0, \frac{M}{p_Y}, 0\right)$ and $\left(0,0, \frac{M}{p_Z}\right)$ and consume the one that yields highest utility.

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  • $\begingroup$ Is this because it will be a corner solution? $\endgroup$ – Brennan Dec 14 '19 at 1:28
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    $\begingroup$ It'll be a corner solution because the utility function is strictly quasi convex. $\endgroup$ – Amit Dec 14 '19 at 1:32

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