I have to find the consumer's demand at prices $\textbf{p} = (1,1,1)$ and budget $m=4$ when the utility function is given by $$u(x_1,x_2,x_3) = 3 \sqrt{x_1+x_2+2x_3}$$ or when transformed $$\hat{u}(x_1,x_2,x_3) = x_1+x_2+2x_3 $$
I know the answer is $\textbf{x} = (0,0,4)$
What I don't understand: For perfect substitutes the demand is given by $x_1^* = \frac{m}{p_1}$ so the demand should have been $\textbf{x} = (4,4,4)$. But why is $x_1^*,x_2^* = 0$?
For the same utility function I want to find the demand at prices $p=(2,3,p_3) >>0$ and budget $m>0$.
The answer is \begin{equation} \textbf{x} = \left(0,0, \frac{m}{p_3} \right) \quad \text{when} \quad p_3<4 \end{equation} \begin{equation} \textbf{x} = \left(\frac{m}{2},0,0 \right) \quad \text{when} \quad p_3>4 \end{equation} \begin{equation} \textbf{x} = \left(\frac{\alpha m}{2}, 0, \frac{(1-\alpha)m}{4} \right) \quad \text{when} \quad p_3=4 \end{equation} where $\alpha \in [0,1]$ because the consumer is indifferent between the bundles
What I don't understand: Why is the demand different depending on what $p_3$ is and how do I know when the demand is different? Personally, I would've said the demand is $\textbf{x} = \left( \frac{m}{2}, \ \frac{m}{3}, \ \frac{m}{p_3} \right)$