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The document you've uploaded contains several economics-related problems involving utility functions, expenditure minimization, and optimal consumption bundles. Here's a brief overview of the problems, which I can help solve step by step:

1. **Consumer's Utility Maximization** (Utility function: \(U(x_1, x_2) = x_1^{1/5} x_2^{4/5}\)):
   - Part (a) asks for the optimal consumption bundle.
   - Part (b) requests the indirect utility function.
   - Parts (c), (d), and (e) explore how the indirect utility function changes with variations in prices and income.

2. **Expenditure Minimization** for the consumer from Problem 1:
   - Find the expenditure function.
   - Show that the expenditure function is homogeneous of degree 1 in prices.
   - Prove that the expenditure function is non-decreasing with respect to prices.

3. **Jaden’s Lunch Problem** (Utility function: \(U(x_1, x_2) = x_1^{1/2} x_2^{1/2}\)):
   - Part (a) asks to find Jaden’s optimal consumption of fried chicken and frozen yogurt.
   - Part (b) explores how much extra money Jaden would need to maintain the same utility if the price of fried chicken increases.

4. **Fredric’s Soda Consumption** (Utility function: \(U(x_1, x_2) = 24\sqrt{x_1} + x_2\)):
   - Part (a) requests Fredric’s optimal consumption bundle.
   - Part (b) asks to calculate the effect of a quantity tax on soda.
   - Part (c) compares the impact of a lump sum tax with a quantity tax.

5. **Tasha’s Nut Consumption** (Utility function: \(U(x_a, x_w) = 3x_a + 5x_w\)):
   - Calculate the maximum membership fee Tasha is willing to pay for a promotion on almonds.

Would you like to work on one of these problems in particular or solve the entire set? Let me know if you want to start with a specific problem or if you'd like detailed explanations of the steps involved!

### 5 related questions:
1. What is an indirect utility function, and how does it relate to the consumer’s preferences?
2. What role does expenditure minimization play in consumer theory?
3. How do price changes impact the consumption choices in utility maximization problems?
4. What is the difference between a quantity tax and a lump sum tax in terms of consumer behavior?
5. How does the concept of homogeneity apply to expenditure functions in economics?

**Tip**: When solving optimization problems in economics, always check if the utility function has diminishing marginal utility and how that affects the consumer’s demand.

A consumer has an income m to spend on two goods. The utility function is given as U(x1, x2) = x1^(1/5) * x2^(4/5). Find the optimal consumption bundle, indirect utility function, and analyze changes due to price and income variations. Solve additional problems on expenditure minimization, and other utility-based consumer decision-making models.

Explore utility maximization and expenditure minimization using various utility functions. Learn to calculate optimal consumption bundles, indirect utility functions, and the impact of price and income changes. This tutorial covers core microeconomic principles including Roy's Identity and Shephard's Lemma, suitable for university-level students.

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