Question: An agent who consumes three commodities has a utility function given by:
$u(x_1,x_2,x_3)=x^{1/3}_1+\min\{ x_2,x_3\}$
Given an income $I$, and prices of $p_1,p_2,p_3$. Describe the consumer’s utility maximization problem. Can the weierstrass and the Kuhn-Tucker theorems be used to obtain and characterize a solution? Why or why not?
attempt: I assume $x_i$ represents the quantity and belongs to $\mathbb R_{+}$. You can form the constraints as follows: $$ x_i \geq0 \quad\forall i \in [3] \\ \sum_{i=1}^3p_ix_i \leq I $$ You can simplify the objective by noting that for the utility to be the maximum. Hence, the final problem becomes,
$$\ \max_{x_1, x_2, x_3}x_1^{1/3} +x_2 \quad s.t. \\ x_i \geq0 \quad\forall i \in [3] \\ \sum_{i=1}^3p_ix_i \leq I \\ x_2=x_3 $$ Let's eliminate $x_3$ as we know that $x_2=x_3$. The problem simplifies to $$\ \min_{x_1,\ x_2}\ -x_1^{1/3} -x_2 \quad s.t. \\ x_1 \geq0,\ x_2 \geq 0 \\ p_1x_1 + (p_2 + p_3)x_2 \leq I $$ $$\ \mathcal L(x_1, x_2)=-x_1^{1/3} -x_2 + \lambda_1(-x_1) + \lambda_2(-x_2) + \lambda(p_1x_1 + (p_2 + p_3)x_2 - I) $$
Comment: I am unsure how to take this further. I keep messing up the derivatives ( I assume) and when I try to solve for lambda I manage to fail to isolate the lambda variable let alone getting the variables x1,x2,x3. My professor encouraged me to try this complex problem as an “exercise for the reader.” How do I carry this further, or can someone show me a step by step solution from this point onwards?