Question

Lan's utility function is U(x,y) =4.9 x³y⁸. If Lan's income is 120 dollars, the price of good X is 2 and the price of good Y is 2, then utility-maximizing amount of good Y is ...

Answer 1

U = 4.9 X³Y⁸

MUx = dU /dX

=> MUx = (4.9) (3) X^3-1 Y⁸.

=> MUx = (4.9) (3) X² Y⁸.

and

MUy = dU / dY

=> MUy = (4.9) (8) X³ Y^8-1.

=> MUy = (4.9) (8) X³ Y⁷.

---------------------

Price of X = 2; Price of Y = 2 and income = 120

Budget constraint:

2X + 2Y = 120

---------------------

At utility maximizing point, (MUx / MUy) = (Price of X / Price of Y)

=> [(4.9) (3) X²Y⁸ / (4.9) (8) X³ Y⁷] = (2 / 2)

=> [3 X2 Y8 / 8 X3 Y7] = 1

=> (3/8) (Y/X) = 1

=> X = (3Y / 8) ----------------------eq(1)

Put eq(1) in budget constraint.

2X + 2Y = 120

=> 2(3Y / 8) + 2Y = 120

=>( 6Y / 8) + 2Y = 120

=> ( 6Y + 2Y *8) / 8 = 120

=> (6Y + 16Y) = 120 *8

=> 22Y = 960

=> Y = (960 / 22)

=> Y = 43.63

The utility maximizing amount of good Y is 43.63