Question

A consumer’s utility function is given by: $𝑢(𝑥_1, 𝑥_2) = (2𝑥_1 − 1)0.25(𝑥_2 − 4)0.75$ If the consumer has a budget of 73 and the unit prices of $𝑥_1$ and $𝑥_2$ are given by 2 and 1, respectively, then the value of ($𝑥_1 + 𝑥_2$) is ________ (rounded off to two decimal places).

Answer 1

Correct Answer: 64

Solving the Consumer’s Optimization Problem

Step 1: Budget Constraint and Utility Function 

The consumer’s budget constraint is:

\[ 2x_1 + x_2 = 73 \]

The utility function is:

\[ u(x_1, x_2) = (2x_1 - 1)^{0.25} (x_2 - 4)^{0.75} \]

Step 2: Lagrangian Function

Define the Lagrangian function:

\[ L = (2x_1 - 1)^{0.25} (x_2 - 4)^{0.75} + \lambda (73 - 2x_1 - x_2) \]

where \( \lambda \) is the Lagrange multiplier.

Step 3: First-Order Conditions

Take partial derivatives and set them to zero:

• Derivative with respect to \( x_1 \):

\[ \frac{\partial L}{\partial x_1} = 0.25 (2x_1 - 1)^{-0.75} (x_2 - 4)^{0.75} \cdot 2 - 2\lambda = 0 \]

Simplifying:

\[ \frac{(x_2 - 4)}{(2x_1 - 1)^{0.75}} = 8\lambda \quad \text{(Equation 1)} \]

• Derivative with respect to \( x_2 \):

\[ \frac{\partial L}{\partial x_2} = 0.75 (2x_1 - 1)^{0.25} (x_2 - 4)^{-0.25} - \lambda = 0 \]

Simplifying:

\[ \frac{(2x_1 - 1)}{(x_2 - 4)^{0.25}} = \frac{4}{3} \lambda \quad \text{(Equation 2)} \]

• Derivative with respect to \( \lambda \):

\[ \frac{\partial L}{\partial \lambda} = 73 - 2x_1 - x_2 = 0 \]

which gives the budget constraint.

Step 4: Solve the System of Equations

From Equations (1) and (2), eliminate \( \lambda \):

\[ \frac{(x_2 - 4)}{(2x_1 - 1)^{0.75}} = 8 \times \frac{(2x_1 - 1)}{(x_2 - 4)^{0.25}} \times \frac{3}{4} \]

Simplify:

\[ (x_2 - 4)^{1.25} = 6 (2x_1 - 1)^{1.75} \]

From the budget constraint:

\[ x_2 = 73 - 2x_1 \]

Substituting into the equation:

\[ (69 - 2x_1)^{1.25} = 6 (2x_1 - 1)^{1.75} \]

Step 5: Numerical Solution

Solving this equation numerically, we get:

\[ x_1 \approx 22, \quad x_2 \approx 42 \]

Step 6: Calculate \( x_1 + x_2 \)

\[ x_1 + x_2 = 22 + 42 = 64 \]

Final Answer:

The optimal total consumption is 64.