Question

Sally the Sleek’s preferences can be described by the utility function U(x, y) = x^2y^3/1000. Prices are px = 4 and py = 3; she has an income of $80 to spend.

(a) If Sally initially consumed 5 units of x and 20 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? (2)

(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 = 5 and y = 20 be optimal? (2)

(c) How much should Sally consume of x and y in order to maximize utility, given her income? (4)

Answer 1

The utility function is given in the question. To find the additional utility, Marginal Utility (MU) needs to be calculated by differentiating the utility function with respect to the good x and y respectively Therefore, the additional utility derived from the good x is 1000 SU 23 δχ 1000 ay 500 (5) (20 500 - 80 the additional utility derived from the good y is 1000 Sy 1000 (5) 3(202 1000 = 30 Therefore, the additional utility derived from good x is S0 and from good y is 30 Initially, the bundle consumed is 5 units of good x and 20 units of good y. If Sally consumed $4 worth more of good x, it means that she consumes 1 more units of good x as the price of good x is $4. Also, the tradeoff between good x and y is 5 units of good x and 20 units of good y. So, when an additional unit of good x must be consumed, it means that Sally needs to give up 4 units of good y. Therefore, the new bundle will be 6 units of good x and 16 units of good y

At bundle (5,20), the utility derived was U(x,y) 1000 (5) (20) 1000 = 200 At bundle (6,16), the utility derived was U(x,y)= 1000 (6) (16) 1000 = 147.46 If she consumed the new bundle, her utility will reduce by $52.55 To be optimal, the following condition should be satisfied MU P MU P 2x 1000 4 1000 2y 4 v-2x The budget constraint is given as 80-4x+31

Solve the above two equations to get the following values: x-8 y 16 Therefore, the optimal bundle is different and thus, the bundle (5,20) is not optimal. Sally should consume 3 units of good and 16 units of good y to maximize utility.