# Optimization question on a function $𝑢(𝑥, 𝑦, 𝑧, 𝑤) = \min\{𝑥, 2𝑦\} + \max \{3𝑧, 4𝑤\}$

I have the following utility function

$$𝑢(𝑥, 𝑦, 𝑧, 𝑤) = \min\{𝑥, 2𝑦\} + \max \{3𝑧, 4𝑤\}$$

I want to find its demand function.

For that

$$\operatorname{Max}𝑢(𝑥, 𝑦, 𝑧, 𝑤) = \min\{𝑥, 2𝑦\} + \max \{3𝑧, 4𝑤\}$$

S.t.

$$p_1 x+p_2 y + p_3 z +p_4 w \le I$$

But I cannot proceed from here. Please suggest a solution. Thanks a lot

Introduce a variable $$v$$ to represent $$\min\{x,2y\}$$ and solve two linear programming problems:
1. Maximize $$v+3z$$ subject to \begin{align} v &\le x \\ v &\le 2y \\ 3z &\ge 4w \\ p_1 x + p_2 y + p_3 z + p_4 w &\le I \\ \end{align}
2. Maximize $$v+4w$$ subject to \begin{align} v &\le x \\ v &\le 2y \\ 3z &\le 4w \\ p_1 x + p_2 y + p_3 z + p_4 w &\le I \\ \end{align}