Suppose that a consumers preferences can be represented by

Question

1. Suppose that a consumer's preferences can be represented by the utility function u(x₁, x₂) = x₁^.9 x₂^.1. If their income is $100, and the prices of goods one and two are $10 each, what fraction of the consumer's income will be spent on good one?

2. Suppose that a consumer has preferences for goods one and two represented by the utility function u(x₁, x₂) = x₁ + x₂. Suppose that the consumer's income is $100 and the price of good two is $10. Plot the consumer's demand curve for good one with the price of good one on the y-axis. Please label at least three points on the graph, and be as clear as possible. Show all your work.

3. Suppose that a consumer's preferences can be represented by the utility function u(x₁, x₂) = x₁^.6 x₂^.4. Suppose that the price of good one, p₁, increases. Use calculus to show what will happen to demand for goods one and two, x₁^* and x₂^*. Are the goods substitutes, complements or neither? Show all your work.

4. Suppose that a consumer's preferences can be represented by the utility function u(x₁, x₂) = ln(x₁) + ln(x₂). Suppose that the consumer's income, m, suddenly increases. Use calculus to show what will happen to demand for goods one and two, x₁^* and x₂^*. For each good determine whether it is an inferior or normal good. Show all your work.

5. Suppose that a consumer's preferences can be represented by the utility function u(x₁, x₂) = min {5x₁, 10x₂}. Suppose that the price of good one, p₁, increases. Use calculus to show what will happen to demand for goods one and two, x₁^* and x₂^*. Are the goods substitutes, complements or neither? Show all your work.

6. Suppose that a consumer's preferences can be represented by the utility function u(x₁, x₂) = min {3x₁, 2x₂}. Suppose that the consumers income, m, suddenly decreases. Use calculus to show what will happen to demand for goods one and two, x₁^* and x₂^*. For each good, determine whether it is an inferior or normal good. Show all your work.

7. Suppose that a consumer's preferences can be represented by the utility function u(x₁, x₂) = x₁^2/3x₂^1/3. Suppose that the originally the price of good one is $3, the price of good two is $3 and the consumer's income is $99. Furthermore, suppose the price of good one then decreases to $1. Calculate the substitution effect and the income effect on good one.

8. Suppose that a consumer's preferences can be represented by the utility function u(x₁, x₂) = min {2x₁, x₂}. Suppose that the originally the price of good one is $2, the price of good two is $2 and the consumer's income is $12. Furthermore, suppose the price of good one then increases to $4. Calculate the substitution effect and the income effect of the price increase on good one.