IGNOU BECC 104 2026 - An individual consumer consumes two commodities X₁ & X₂. The utili

Question

An individual consumer consumes two commodities X₁ & X₂. The utility function is

$$ U = X_{1}^{0.4} X_{2}^{0.6}$
The price of commodity one is P₁ = Rs.3.00, the price of commodity two is P₂ = Rs.4.00, the individual's income per period is Rs.108. Determine the utility maximizing level of X₁ & X₂ and derive the demand curves for the two commodities.

25 Mar 2026

Answer :

Word Count : 362

The consumer maximizes utility subject to the budget constraint. The problem is: Utility function: [ U = X_1^{0.4} X_2^{0.6} ] Budget constraint: [ P_1 X_1 + P_2 X_2 = I \quad \Rightarrow \quad 3X_1 + 4X_2 = 108 ] Step 1: Set up the Lagrangian: [ \mathcal{L} = X_1^{0.4} X_2^{0.6} + \lambda (108 - 3X_1 - 4X_2) ] Step 2: First-order conditions (FOC) by partial derivatives: [ \frac{\partial \mathcal{L}}{\partial X_1} = 0.4 X_1^{-0.6} X_2^{0.6} - 3\lambda = 0 \quad \Rightarrow \quad \lambda = \frac{0.4 X_1^{-0.6} X_2^{0.6}}{3} ] [ \frac{\partial \mathcal{L}}{\partial X_2} = 0.6 X_1^{0.4} X_2^{-0.4} - 4\lambda = 0 \quad \Rightarrow \quad \lambda = \frac{0.6 X_1^{0.4} X_2^{-0.4}}{4} ] Step 3: Equate the two expressions for (\lambda): [ \frac{0.4 X_1^{-0.6} X_2^{0.6}}{3} = \frac{0.6 X_1^{0.4} X_2^{-0.4}}{4} ] Simplify step by step: Multiply both sides by 12 to eliminate denominators: [

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