Question

Suppose utility function of a consumer is \( u(x, y) = xy \), where \( x \) and \( y \) are quantities of the two commodities consumed. If price of \( x \) is Rs.1 and that of \( y \) is Rs.2, and income of the consumer is Rs.100, utility maximizing quantity consumption of \( x \) is ..........

Hint:

To maximize utility, set the ratio of the marginal utilities equal to the price ratio, then use the budget constraint to solve for the quantities of each good.

Answer 1

The consumer’s utility function is \( u(x, y) = xy \).
Step 1: Budget constraint is given by: \[ P_x x + P_y y = I \] Where:
- \( P_x = 1 \) (price of \( x \)),
- \( P_y = 2 \) (price of \( y \)),
- \( I = 100 \) (income of the consumer).
Thus, the budget constraint is: \[ x + 2y = 100 \] Step 2: The consumer maximizes utility by choosing \( x \) and \( y \) such that the ratio of marginal utilities equals the price ratio: \[ \frac{MU_x}{MU_y} = \frac{P_x}{P_y} \] Where the marginal utilities are the partial derivatives of the utility function: \[ MU_x = \frac{\partial u}{\partial x} = y \quad \text{and} \quad MU_y = \frac{\partial u}{\partial y} = x \] Step 3: Setting the ratio of marginal utilities equal to the price ratio: \[ \frac{y}{x} = \frac{1}{2} \] This implies: \[ y = \frac{x}{2} \] Step 4: Substituting \( y = \frac{x}{2} \) into the budget constraint: \[ x + 2\left( \frac{x}{2} \right) = 100 \] \[ x + x = 100 \] \[ 2x = 100 \] \[ x = 50 \] Thus, the utility-maximizing quantity of \( x \) is 50. Final Answer: \[ \boxed{50} \]