The consumer's problem, as outlined in the given Lagrangian function $L = U(x, y) + \lambda(I - P_x x - P_y y)$, is to maximize their utility subject to the budget constraint $P_x x + P_y y = I$.

Correct Option:

Max $U(x, y)$ s.t. $P_x x + P_y y = I$.

Explanation:

1. Objective: The consumer wants to maximize utility $U(x, y)$, which depends on the quantities of goods $x$ and $y$ consumed.
2. Constraint: The budget constraint ensures that the total expenditure on goods $x$ and $y$, $P_x x + P_y y$, does not exceed their income $I$.
3. Method: The Lagrangian approach is used to incorporate the constraint into the optimization problem using the multiplier $\lambda$.

Let me know if you want me to solve this further or clarify the steps! Below are some related questions.

Related Questions:

1. What are the first-order conditions (FOCs) for this optimization problem?
2. How does the Lagrange multiplier $\lambda$ relate to the marginal utility of income?
3. What happens if the prices $P_x$ or $P_y$ increase? How does it affect consumption?
4. Can you explain the significance of the marginal rate of substitution in this context?
5. What would be the dual problem associated with this consumer's maximization problem?

Tip:

To solve Lagrangian problems, always set up the first-order conditions systematically, as they provide the solution to both the decision variables and the constraint simultaneously.