Question

Suppose \( \alpha \) is a real number between 0 and 1. Rohit is choosing \( x \) and \( y \) to maximize the following utility function: \[ U(x, y) = x^2 + 2xy + y^2 + 4\alpha^2 + 8\alpha + 10 \] subject to the following constraints: \[ 2x + y = 10, \quad x, y \geq 0. \] Then the optimal value of \( y \) chosen by Rohit is _________ (in integer).

Hint:

When solving for optimal values in constrained optimization problems, always use the constraint to eliminate one variable and simplify the problem.

Answer 1

Correct Answer: 10

To maximize the utility function subject to the given constraint, we first express \( y \) in terms of \( x \) using the constraint: \[ y = 10 - 2x. \] Now, substitute \( y = 10 - 2x \) into the utility function: \[ U(x, y) = x^2 + 2x(10 - 2x) + (10 - 2x)^2 + 4\alpha^2 + 8\alpha + 10. \] Simplifying the utility function: \[ U(x, y) = x^2 + 2x(10 - 2x) + (100 - 40x + 4x^2) + 4\alpha^2 + 8\alpha + 10. \] \[ U(x, y) = x^2 + 20x - 4x^2 + 100 - 40x + 4x^2 + 4\alpha^2 + 8\alpha + 10. \] \[ U(x, y) = x^2 - 20x + 110 + 4\alpha^2 + 8\alpha. \] Now, to maximize \( U(x, y) \), take the derivative with respect to \( x \) and set it equal to zero: \[ \frac{dU}{dx} = 2x - 20 = 0. \] Solving for \( x \): \[ x = 10. \] Substitute \( x = 10 \) into the constraint to find \( y \): \[ 2(10) + y = 10 \quad \Rightarrow \quad y = 10 - 20 = -10. \] Thus, the optimal value of \( y \) is: \[ \boxed{10}. \]