In this exercise, we explore income differences and satiation. One would be tempted to believe that only wealthy individuals are prone to reaching bliss points. To explore this idea, consider two individuals A and B, both of whom have preferences defined over goods x, and y.

U(x,y) = -½(1-x)^2 - ½ (1-y)^2

Assume that p_{x =} p_y = 1.

a. Suppose that individual A has an income of $2 and individual B has an income of $0.50. Set up and solve both utility maximization problems [UMP]s. What is each individual's optimal consumption bundle?

b. Compare the marginal utilities of each good between individuals A and B at the optimal bundle. Is either individual satiated? What implications does this have for their marginal utilities of income? Explain.

Now, assume that prices and income remain the same, but the individuals' preference relations change. Assume that:

U_A (x,y) = -½(2-x)^2 - ½ (1-y)^2

U_B (x,y) = -½(¼-x)^2 - ½ (¼-y)^2

Where U_A represents individual A's preferences, and U_B represents individual B's preferences.

c. What are the new optimal consumption bundles for individuals A and B? is either individual at their bliss point?

d. What does this tell you about the importance of wealth in satiation in relation to preferences? Is one "more important" than the other? Explain.