Edgar's utility function is: U(x, y) = xy5

1) What is Edgar's demand function for good X? [This should be an equation for X that contains only Px, Py, and/or M.] Please write your answer as "X* = ". You can use lowercase letters for the subscripts. Make sure the equation is simplified.


To find Edgar's demand function for good \( X \), we start with his utility function and assume that he maximizes utility subject to his budget constraint.

### Step 1: Define the Utility Function and Budget Constraint

1. **Utility function**: \( U(x, y) = x \cdot y^5 \)
2. **Budget constraint**: \( p_x \cdot x + p_y \cdot y = M \), where \( p_x \) and \( p_y \) are the prices of goods \( X \) and \( Y \), respectively, and \( M \) is Edgar's income.

### Step 2: Set Up the Lagrangian

To maximize utility given the budget constraint, we can set up the Lagrangian:
\[
\mathcal{L} = x \cdot y^5 + \lambda (M - p_x \cdot x - p_y \cdot y)
\]

### Step 3: Solve the First-Order Conditions

1. **Partial derivative with respect to \( x \):**
   \[
   \frac{\partial \mathcal{L}}{\partial x} = y^5 - \lambda p_x = 0 \Rightarrow \lambda = \frac{y^5}{p_x}
   \]

2. **Partial derivative with respect to \( y \):**
   \[
   \frac{\partial \mathcal{L}}{\partial y} = 5x \cdot y^4 - \lambda p_y = 0 \Rightarrow \lambda = \frac{5x \cdot y^4}{p_y}
   \]

3. **Partial derivative with respect to \( \lambda \):**
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda} = M - p_x \cdot x - p_y \cdot y = 0
   \]

### Step 4: Equate the First-Order Conditions

From the two expressions for \( \lambda \), we have:
\[
\frac{y^5}{p_x} = \frac{5x \cdot y^4}{p_y}
\]

Solving for \( x \) in terms of \( y \), \( p_x \), and \( p_y \):
\[
y^5 \cdot p_y = 5x \cdot y^4 \cdot p_x
\]
\[
y \cdot p_y = 5x \cdot p_x
\]
\[
x = \frac{y \cdot p_y}{5p_x}
\]

### Step 5: Substitute \( x \) in the Budget Constraint

Using \( x = \frac{y \cdot p_y}{5p_x} \) in the budget constraint:
\[
p_x \cdot \frac{y \cdot p_y}{5p_x} + p_y \cdot y = M
\]
\[
\frac{y \cdot p_y}{5} + p_y \cdot y = M
\]
\[
p_y \cdot y \left(1 + \frac{1}{5}\right) = M
\]
\[
\frac{6p_y \cdot y}{5} = M
\]
\[
y = \frac{5M}{6p_y}
\]

### Step 6: Substitute \( y \) Back to Find \( x \)

Now substitute \( y = \frac{5M}{6p_y} \) into \( x = \frac{y \cdot p_y}{5p_x} \):
\[
x = \frac{\left(\frac{5M}{6p_y}\right) \cdot p_y}{5p_x}
\]
\[
x = \frac{5M \cdot p_y}{6p_y \cdot 5p_x}
\]
\[
x = \frac{M}{6p_x}
\]

### Answer

Thus, Edgar's demand function for good \( X \) is:
\[
X^* = \frac{M}{6p_x}
\]

Would you like more details on any step?

Here are five related questions to explore:

1. How would Edgar's demand function for \( Y \) look?
2. What happens to Edgar’s demand for \( X \) if his income \( M \) doubles?
3. How does a price increase in \( p_x \) affect Edgar's utility maximization?
4. If the utility function changes to \( U(x, y) = x^2 y^3 \), how would the demand function change?
5. Can we derive the indirect utility function given the demand functions for \( X \) and \( Y \)?

**Tip**: Remember, the Lagrangian method is a powerful way to maximize a utility function with constraints, especially for functions with multiple variables and a budget constraint.

Edgar's utility function is: U(x, y) = xy^5

1) What is Edgar's demand function for good X? [This should be an equation for X that contains only Px, Py, and/or M.] Please write your answer as "X* = ". You can use lowercase letters for the subscripts. Make sure the equation is simplified.

This page explains how to derive Edgar's demand function for good X given his utility function U(x, y) = xy^5 and a budget constraint. The step-by-step solution covers utility maximization, the Lagrangian method, and first-order conditions, suitable for undergraduate economics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation