Sally the Sleek’s preferences can be described by the utility function U(x,y) = x^2y^3/512. Prices are px = 2 and py = 6; she has an income of $80 to spend.
(a) If Sally initially consumed 4 units of x and 12 units of y, how much additional utility does she get from spending one (small fraction of a) dollar more on good x? How much additional utility does she get from spending one (small fraction of a) dollar more on good y?
(b) By how much would her utility change if she stayed on the same budget and consumed 4 (“small”) dollars worth more of x? Judging from this, can the allocation of x = 4 and y = 12 be optimal?
(c) How much should Sally consume of x and y in order to maximize utility, given her income?