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To tackle part (b), we need to derive a utility function from the expenditure function \( e(p,u) \) such that it generates the demand functions given in Example 2.3.

1. **Given Expenditure Function**: 
   \[
   e(p, u) = u p_1^{\alpha_1} p_2^{\alpha_2} p_3^{\alpha_3}
   \]
   where \( p = (p_1, p_2, p_3) \) represents the vector of prices and \( u \) is the utility level.

2. **Constructing the Utility Function**:
   - We need to use the formula from the proof of Theorem 2.1, which is generally used to invert the expenditure function to get the indirect utility function \( v(p, y) \), where \( y \) is the income.
   - The indirect utility function \( v(p, y) \) can be obtained by solving \( e(p, u) = y \) for \( u \):
   \[
   u = \frac{y}{p_1^{\alpha_1} p_2^{\alpha_2} p_3^{\alpha_3}}
   \]

3. **Utility Function**:
   - Now, from the indirect utility function \( v(p, y) \), the direct utility function \( U(x_1, x_2, x_3) \) can be derived. To recover the utility function \( U \), we need to find the demand functions \( x_i \) that solve the utility maximization problem subject to the budget constraint.
   - The demand functions for Cobb-Douglas preferences generally take the form:
   \[
   x_i = \frac{\alpha_i y}{p_i}
   \]
   where \( \alpha_i \) represents the share of income spent on good \( i \).

4. **Verification**:
   - To verify that the derived utility function generates the demand functions in Example 2.3, you would substitute these demand functions back into the utility function form and check if the functional form holds true.

5. **Conclusion**:
   - By substituting the derived demand functions into the utility function, if it matches the form in Example 2.3, we have successfully shown the utility function generates the given demand functions.

Would you like to see the detailed mathematical derivations, or do you have any specific parts you would like to explore further?

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**5 Related Questions**:
1. How is the expenditure function used to derive the indirect utility function?
2. What is the relationship between the indirect utility function and the demand functions?
3. How does Roy's identity connect the indirect utility function to demand functions?
4. Can you explain the proof of Theorem 2.1 in detail?
5. How can the Cobb-Douglas demand functions be derived from the utility maximization problem?

**Tip**: For verifying derivations, always plug the results back into the original functions or equations to ensure consistency and correctness.

Derive a utility function from the expenditure function e(p, u) and verify Roy's identity.

Learn how to derive the indirect utility function from the expenditure function e(p, u) and verify Roy's identity. This involves key economic concepts such as demand functions and utility maximization. Suitable for undergraduate economics or mathematics students.

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