2 A household decision problem A couple has two young children, Dillon and Jessie, who spend their time playing with blocks and dolls. Each child prefers to have more blocks and dolls to play with, and their total utilities are UD($b, 33d) = 53b + 233d: UJ(Eb, 93d) = 3:85 + 113.1: where .735 is the number of blocks in the house, and an; is the number of dolls. (Throughout this problem, assume that blocks and dolls can be purchased in fractional amounts.) Parents get only as much free time as their least happy child allows them, and so their utility is the minimum of that of Dillon and Jessie: Up($b,$d) = Inj{UD(L'b,$d), UJ($b,xd)}. 1. Plot the set of kinks of Up. 2. On the same graph, plot two indifference curves of U p, and shade in an upper contour set. Is the underlying preference relation convex? Is it monotone? 3. Suppose that dolls cost $10 each while blocks cost 133, and the parents have budgeted $120 to spend on toys. For each of the following values of pa, determine what toys the parents will buy. Illustrate each solution on a new graph by drawing the budget constraint, the optimal bundle, and the indifference curve through that bundle. 0 p3 = $10 0 p3 = $40 P3=$2