Thew has a utility function U= (xy)^1/2  (where Y represents apples and X represents hot dogs), income of $20, and is deciding how to allocate that income between apples and hot dogs. Hot dogs cost $1.20 each while apples cost $1.00 each.

i) Matthews utility function implies that the marginal utility of apples is 0.5 (xy)  and the marginal utility of hot dogs is 0.5 (yx). How many apples will Matthew buy? How many hot dogs will he buy?

ii) This year, the price of hotdogs rise to $3 each while Matthew's income is unchanged. Matthew's father decides to help him by giving him a gift of $20. Consider an indifference curve-budget line diagram with hot dogs on the x-axis and apples on the y-axis. After Matthew gets his $20 gift, will his new budget line lie above, lie below, or pass through his initial optimum? Justify your choice. Will Matthew be better or worse off than he was last year?