Problem 3. Suppose David spends his income (I) on two goods, x and y, whose market prices are px and py, respectively. His preferences are represented by the utility function u(x, y) = lnx + 2lny. (MUx = 1/x, MUy = 2/y) (a) Derive his demand functions for x and y. (b) Assuming px = $1 and py = $2, graph his Engel curve for x. (c) Assuming I = $60 and px = $1, graph his demand curve for y. (d) From the result of (a), is good x normal good or inferior good? Explain why.
Problem 4. Consider three consumers with the respective utility functions U A(x, y) = √xy (MRS(x,y)=y/x) U B(x, y) = x + y (perfect substitutes) U c (x, y) = min(x, y) (perfect complements) (a) a. Assume each consumer has income $120 and initially faces the prices px = $1 and py = $2. How much x and y would they each buy? (b) Next, suppose the price of x were to increase to $4. How much would they each buy now? (c) Decompose the total effect of the price change on demand for x into the substitution effect and the income effect. That is, determine how much of the change is due to each of the component effects. (Hint 1: For agent A, what two properties determine the location of z, the reference point for distinguishing the income and substitution effects? Hint 2: For agents B and C, identify the substitution effect, i.e., the point z, graphically by moving the new budget line up until it just touches the original indifference curve.).