How to Maximize Utility Subject to Budget Constraint Using Lagrangian Method
What is the Lagrangian function for maximizing utility subject to the budget constraint?
If you want to maximize Utility subject to the budget constraint, the Lagrangian would be specified as:
A. L = 4x1^(1/2) x2 - (8x1 + 12x2)
B. L = 12x1 + 8x2 - 96 - λ (4x1^(1/2) x2)
C. L = 4x1^(1/2) x2 + 96 - (8x1 + 12x2)
D. L = 4x1^(1/2) x2 - (8x1 + 12x2 - 96)
Answer:
The Lagrangian for maximizing utility subject to the budget constraint in this case would be option D: L = 4x₁^(1/2)x₂ - (8x₁ + 12x₂ - 96).
To maximize utility subject to the budget constraint, we can use the Lagrangian method. The Lagrangian function involves the utility function and the budget constraint, along with a Lagrange multiplier (λ) to incorporate the constraint.
In this case, the utility function is U = 4x₁^(1/2)x₂, where x₁ and x₂ are the quantities of goods 1 and 2, respectively. The prices of goods 1 and 2 are given as P₁ = 8 and P₂ = 12, and the income (m) is 96.
The budget constraint is represented as P₁x₁ + P₂x₂ = m, which translates to 8x₁ + 12x₂ = 96.
To construct the Lagrangian, we subtract the budget constraint from the utility function multiplied by the Lagrange multiplier:
L = 4x₁^(1/2)x₂ - (8x₁ + 12x₂ - 96)
Therefore, the correct Lagrangian in this case is option D: L = 4x₁^(1/2)x₂ - (8x₁ + 12x₂ - 96).