Question

Suppose a consumer’s preferences are represented by the utility function

U(X,Y) = X²*Y.

Therefore,

MU_x = 2XY
•    MU_y = X²

Also, suppose the consumer has $32 to spend (M = $32), P_Y = 1, and that they spend all of their money on goods X and Y. Also, assume the consumer maximizes their utility subject to their budget constraint. Complete the following table:

Px	Quantity Demanded of X
$1
$2
$3

Answer 1

U(X,Y) = X²Y  

MU_X = 2XY  

MU_Y = X²  

MRS =  MU_X /MU_Y

= 2XY/X²  

= 2Y/X  

At utility maximizing point   

MRS = Px/Py  

2Y/X = Px/Py  

2Y = (Px/Py)X

Y = (1/2)(Px/Py)X

BUDGET LINE

PxX + PyY = M

Put    Y = (1/2)(Px/Py)X in BL

PxX + PyY = M

PxX + (1/2)Py(Px/Py)X = M

(2PxX + PxX)/2 = M

3PxX = 2M

X = 2M/3Px

when M = 32

Py = 1

Px = 1

X = 2M/Px

= 2(32)/(3)(1)

= 64/3

When M = 32

Py = 1

Px = 2

X = 2(32)/(3)(2)

= 64/6

= 32/3

When M = 32

Py = 1

Px = 3

X = 2(32)/(3)(3)

= 64/9