The question involves solving a utility maximization problem using the Lagrangian multiplier technique.

Solution for Part (a):

The utility function is given by: $U(X_1, X_2) = X_1^{0.25} X_2^{0.75}$

Prices of commodities: $P_{X_1} = 5 \, \text{Ksh/unit}, \, P_{X_2} = 10 \, \text{Ksh/unit}$

Income: $I = 30,000 \, \text{Ksh}$

Steps to Maximize Utility:

1. Budget Constraint: $P_{X_1} X_1 + P_{X_2} X_2 = I$ Substituting values: $5X_1 + 10X_2 = 30,000$

2. Lagrangian Function: $\mathcal{L} = X_1^{0.25} X_2^{0.75} + \lambda (30,000 - 5X_1 - 10X_2)$

3. Partial Derivatives: Take partial derivatives w.r.t $X_1$, $X_2$, and $\lambda$: $\frac{\partial \mathcal{L}}{\partial X_1} = 0.25X_1^{-0.75} X_2^{0.75} - 5\lambda = 0$ $\frac{\partial \mathcal{L}}{\partial X_2} = 0.75X_1^{0.25} X_2^{-0.25} - 10\lambda = 0$ $\frac{\partial \mathcal{L}}{\partial \lambda} = 30,000 - 5X_1 - 10X_2 = 0$

4. Solving the Equations:

   • From the first two equations: $\frac{0.25X_1^{-0.75} X_2^{0.75}}{5} = \frac{0.75X_1^{0.25} X_2^{-0.25}}{10}$ Simplify: $\frac{X_2}{X_1} = 1.5$ Hence: $X_2 = 1.5X_1$

   • Substitute $X_2 = 1.5X_1$ into the budget constraint: $5X_1 + 10(1.5X_1) = 30,000$ $5X_1 + 15X_1 = 30,000$ $20X_1 = 30,000 \implies X_1 = 1,500$

   • Calculate $X_2$: $X_2 = 1.5 \times 1,500 = 2,250$

Final Results:

• Optimal quantities: $X_1 = 1,500, \, X_2 = 2,250$

Explanation for Part (b):

Using a table, describe and explain practical applications of the Cardinal Utility Approach. This involves scenarios like consumer decision-making under budget constraints, pricing strategies, and market analysis.

Would you like detailed explanations for part (b) or more elaboration on the mathematical derivations?

Related Questions:

1. What is the significance of the Lagrangian multiplier in this context?
2. How would the solution change if the income were doubled?
3. Can you derive the marginal rate of substitution (MRS) at the optimal point?
4. What are the practical implications of this optimization problem in market economics?
5. How does the elasticity of substitution between $X_1$ and $X_2$ influence the utility maximization?

Tip:

Always check the validity of your solutions by verifying they satisfy both the utility function and the budget