Problem 2 A consumer has $40 to spend on xylophones (x) and yardsticks (y). Each xylophone costs $5 and each yardstick costs $8. The consumer derives material welfare from these two goods according to the following utility function: u(x, y) = 4x + 7y. Non-integer units of consumption are allowed (even though these two items are naturally supposed to be purchased in discrete units). In this problem, assume that the consumer is not required to spend all of her income, and do impose the non-negativity constraints on consumption explicitly. Use the Kuhn - Tucker conditions to prove that, in optimum, (i) the consumer purchases only yardsticks and no xylophones; and (ii) the consumer spends all of her income.