https://file.aiquickdraw.com/mathbot/file/1727339515545khkmghtl.jpg

Let's address both problems from the image:

### Problem 2.3:
We are asked to derive the consumer’s **direct utility function** from the given **indirect utility function**:

\[
v(p, y) = y p_1^\alpha p_2^\beta
\]
where \( p = (p_1, p_2) \) and \( \alpha, \beta \) are negative.

To derive the **direct utility function**, we need to use **Roy's Identity**. Roy's identity relates the **indirect utility function** \( v(p, y) \) to the **Marshallian demand functions** \( x_1(p, y) \) and \( x_2(p, y) \) as follows:

\[
x_i(p, y) = -\frac{\partial v(p, y) / \partial p_i}{\partial v(p, y) / \partial y}
\]

1. **Step 1**: Find the partial derivatives of the indirect utility function \( v(p, y) \) with respect to \( p_1 \), \( p_2 \), and \( y \).
2. **Step 2**: Apply Roy's identity to derive the demand functions.
3. **Step 3**: Use the demand functions to recover the **direct utility function**.

### Problem 2.5 (Part a):
We are given the **expenditure function**:

\[
e(p, u) = u p_1^{\alpha_1} p_2^{\alpha_2} p_3^{\alpha_3}
\]
We are asked to:
- Derive the **indirect utility function** using the relation \( e(p, v(p, y)) = y \).
- Verify **Roy's identity**.

Here, you will:
1. **Step 1**: Invert the expenditure function to express \( u \) in terms of \( p \) and \( y \) (this gives the indirect utility function).
2. **Step 2**: Apply Roy's identity to the resulting indirect utility function to verify consistency.

### Problem 2.5 (Part b):
You are required to reconstruct the **direct utility function** from the expenditure function \( e(p, u) \) using a method similar to Theorem 2.1. Specifically, verify that the resulting utility function generates the demand functions given in **Example 2.3**.

This involves:
1. **Step 1**: Use **dual relationships** between the expenditure function and the utility function.
2. **Step 2**: Differentiate and confirm that the demand functions are correct.

Would you like to focus on solving one of these problems step by step in detail, starting with the derivation of the direct utility function for problem 2.3?

### Related Questions:
1. How is Roy's identity generally derived in consumer theory?
2. How does the expenditure function relate to indirect utility in duality theory?
3. What are the properties of indirect utility functions when prices are variable?
4. Can the demand functions derived from indirect utility functions always lead to a consistent direct utility function?
5. How does the assumption of negative \( \alpha \) and \( \beta \) affect the form of the utility functions?

#### Tip:
Remember, duality in microeconomics allows for switching between utility maximization (direct utility) and expenditure minimization (indirect utility) problems efficiently. Understanding this principle helps in working with utility and demand functions.

Derive the consumer's direct utility function if the indirect utility function has the form v(p, y) = y p_1^α p_2^β for negative α and β. Also, consider the solution e(p,u) = u p_1^α_1 p_2^α_2 p_3^α_3 and perform the tasks (a) and (b).

Learn how to derive a consumer's direct utility function from the given indirect utility function v(p, y) = y p_1^α p_2^β for negative α and β using Roy's identity. Explore related problems, including deriving the indirect utility function using the expenditure function and verifying consistency with Roy's Identity.

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