Question

Trisha’s consumption preference on biryani \( (x) \) and pudding \( (y) \) is given by the utility function \( U(x, y) = x + 4y \). The price per unit of biryani is ₹2 and the price per unit of pudding is ₹3. Trisha’s total income is ₹120. However, she faces an extra quantity constraint as she is not allowed to consume biryani more than 60 units and pudding more than 30 units. The optimum quantity of biryani and pudding consumed by Trisha is

• A. \( (x^, y^) = (30, 20) \)
• B. \( (x^, y^) = (15, 30) \)
• C. \( (x^, y^) = (30, 15) \)
• D. \( (x^, y^) = (60, 0) \)

Hint:

When faced with a utility maximization problem, use the budget constraint to express one variable in terms of the other and substitute it into the utility function.

Answer 1

The Correct Option is B

The total expenditure is \( 2x + 3y \), and Trisha’s total income is ₹120, so: \[ 2x + 3y = 120. \] We also have the constraints \( x \leq 60 \) and \( y \leq 30 \). To maximize her utility function \( U(x, y) = x + 4y \), we solve for \( y \) in terms of \( x \): \[ y = \frac{120 - 2x}{3}. \] Now, substitute this into the utility function: \[ U(x) = x + 4\left(\frac{120 - 2x}{3}\right) = x + \frac{480 - 8x}{3}. \] Simplifying: \[ U(x) = \frac{3x + 480 - 8x}{3} = \frac{-5x + 480}{3}. \] To maximize utility, we differentiate with respect to \( x \) and set it equal to zero: \[ \frac{dU}{dx} = \frac{-5}{3} = 0. \] This is a maximum at \( x = 30 \). Substituting \( x = 30 \) into the budget constraint: \[ 2(30) + 3y = 120 \quad \Rightarrow \quad 60 + 3y = 120 \quad \Rightarrow \quad y = 20. \] Thus, the optimum quantities are \( x^ = 30 \) and \( y^ = 20 \). Final Answer: \boxed{(x^, y^) = (30, 20)}