Lucy consumes only scoops of ice cream (x) and cones (y). Moreover, she insists on consuming these two goods in the combination of 1 scoop of ice cream and 1 cone. If there are more scoops of ice cream than cones, she throws the extra ice cream away. If there are more cones than scoops of ice cream, she throws the extra cones away. Her utility function of ice cream and cones is given by U(x, y) = min {x, y}. (The function min {x, y} returns the smaller number between x and y. In other words, if x < y, min {x, y} = x; if x > y, min {x, y} = y; and if x = y, min {x, y} = x = y.)

(a) Draw a couple of indifference curves for Lucy. Put ice cream on the horizontal axis. (Hint: Draw the 45 line. Choose a point on the line. If you increase the amount of x but not y from that point, how would Lucy's utility change? If you increase the amount of y but not x from that point, how would Lucy's utility change?)

(b) Suppose each scoop of ice cream costs $2, and each cone costs $1. Lucy has an income of $6. Draw her budget line in the diagram you have drawn in part (a).

(c) Using the diagram you have drawn, find Lucy's utility-maximizing consumption bundle. Lucy's sister, Mary, also insists on a particular combination of ice cream and cones. Unlike Lucy, however, Mary must have 2 scoops of ice cream with every cone. Mary's utility function is given by U(x, y) = min x 2 , y .

(d) In a separate diagram, draw a couple of indifference curves for Mary. (Hint: Instead of drawing the 45 line, draw the line y = x/2. Then follow the procedures as in part (a).)

(e) Mary faces the same prices as Lucy but has an income of $10 instead. Using the diagram you have drawn in part (d), find Mary's utility maximizing consumption bundle.