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

Question

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

Answer 1

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}

Take whichever solution yields the larger of the two resulting objective values.