# How to find Marshallian demand of $u(x,y,z)=x+y^2+2z^2$?

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

• 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. – Brennan Dec 13 '19 at 2:04
• @Brennan Does the convexity of this function not matter at all, are you sure first order conditions are sufficient? – Giskard Dec 13 '19 at 5:36
• That's why I'm asking, I can't get far with the usual interior conditions. – econ86 Dec 13 '19 at 9:02
• @Giskard very good point that I overlooked – 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.